Warm Up State the slope and the y-intercept for each of the equations below:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Warm Up State the slope and the y-intercept for each of the equations below:"

Transcription

1 Warm Up State the slope and the y-intercept for each of the equations below: 1) y = 3x + 2 2) y = -2x + 6 3) 2x + 3y = 15 4) 6y - 5x = -12

2 A SYSTEM OF EQUATIONS is a set of two or more equations that contain two or more variables. A SOLUTION OF A SYSTEM OF EQUATIONS is the point of intersection of the lines or can be said to be a set of values that are solutions of all the equations.

3 If you can graph a straight line, you can solve systems of equations graphically! Solving a system by graphing is very easy!!! To solve a system of equations graphically, just graph both equations and see where they intersect (cross). The point of intersection is the solution. FOR EXAMPLE: 4x + y = 8 y - x = 3 Graph: y = -4x + 8 y = x + 3 Point of intersection is:

4 Solutions to Systems of Equations There are 3 possible solutions for a system of equations: 1) 1 solution - this happens when the graphs of the equations intersect at exactly 1 point. y =x 4 y= 2x +5 Called consistent and independent. Notice the lines have different slopes! Solutions to Systems of Equations 2) No solution - this happens when the graphs of the equations never intersect, which means they are parallel lines. y = -x + 3 y = -x - 1 Called inconsistent. Notice the lines have the same slope BUT different y intercepts.

5 Solutions to Systems of Equations 3) Infinte solution - this happens when the graphs of the equations are the same line. This means they have the same slope and y- intercept. 2x + y = 3 4x + 2y = 6 Called consistent and dependent. Notice the lines have the same slope AND the same y intercepts. (Put into y=mx+b form) Solutions to Systems of Equations Description of Graph Number of Solutions Special Terminology Intersecting Lines Exactly One Consistent & Independent Parallel Lines NONE Inconsistent Same Line (Coincidental) Infinitely Many Consistent & Dependent

6 = Let's try another one: You try this one: =

7 Practice on Solving Systems by Graphing Solve each system by graphing: 1) y = 3x -4 y = -3x + 2 2) y = 4/3 x + 3 y = 2/3 x 3 3) y = 5/4 x -2 y = 5/4 x - 1 Practice on Solving Systems by Graphing Solve each system by graphing: 1) y = 3x -4 y = -3x + 2 2) y = 4/3 x + 3 y = 2/3 x 3 3) y = 5/4 x -2 y = 5/4 x - 1 ( 3, 1) no Solution (1, 1)

8 Graph the following and find the point of intersection, or solution ANSWERS 1) (-4, 7) 2) (3, -1) 3) (0, 1) 4) (3, 6) 5) (0, 2) 6) (0, -3)

9 Homework: Textbook pg 610 #1-6, 8, 10, 14, 16, 20 The substitution method is used to eliminate one of the variables by replacement when solving a system of equations! Think of it as "grabbing" what one variable equals from one equation and "plugging" it into the other equation.

10 STEPS FOR SOLVING A SYSTEM OF EQUATIONS BY SUBSTITUTION 1) Solve one of the equations for either "x =" or "y =". 2) Replace the "y" value in the first equation by what "y" now equals. Solve this new equation for "x". 3) Place this new "x" value into either of the ORIGINAL Pick the easier one to work with! 4) CHECK: Substitute your "x" and "y" values into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE! FOR EXAMPLE: 3y - 2x = 11 y + 2x = 9

11 3x + 2y = 10 2x - y = 9 Let's try another system: 4x - y = 6 -x + y =3

12 You try this one: x + 4y = -10 x - 3y = 11 SPECIAL CASES We know the solution to a system is the intersection of two lines. Let's look at two special cases. CASE 1: y = 2x + 9 2x - y = -2 The solution to this system is

13 CASE 2: 10y = 2x - 8 x - 5y = 4 The solution to this system is Here is what you should remember! SYSTEMS OF EQUATIONS CAN HAVE... a POINT OF INTERSECTION which will be the answer to the system, no intersection (because they are parallel) and the answer to the system is NO SOLUTION, or many points of intersection (because they are the same line) and the answer to the system is INFINITELY MANY SOLUTIONS.

14 Look at the following examples: 6x - y = 12-12x + 2y = 4 x + y = 1-2y = 2x - 2 HOMEWORK Do the Pizzazz worksheet "Why Does the President Put Vegetables in His Blender?"

15 Simultaneous equations got you baffled? RELAX! You can do it! Think of the adding or subtracting method as simply "eliminating" one of the variables to make your life easier. STEPS FOR SOLVING A SYSTEM OF EQUATIONS BY ELIMINATION 1) First, "line" the variables up under one another. 2) Decide which variable ("x" or "y") will be easier to eliminate. In order to eliminate a variable, the numbers in front of them (the coefficients) must be negatives of one another. 3) Now, subtract to eliminate the "x" or "y" variable. 4) Solve the simple equation. 5) Plug into either of the ORIGINAL equations to get the value of the other variable. Pick the easier one to work with! 6) CHECK: Substitute your "x" and "y" values into BOTH ORIGINAL equations. If these answers are correct, BOTH equations will be TRUE!

16 FOR EXAMPLE: 9x -3y = 3 3x + 8y = -17 Let's try another system: 2x + 4y = -4 3x + 5y = -3

17 Let's try another system: 5x + 2y = -1 3x + 7y = 11 You try this one: -2x + 5y = -17 3x - 10y = 28-5x + 2y = 32 2x + 3y = 10

18 You try this one: 3x + 2y = 6-6x - 4y = -12 2x - 4y = -1 6x - 12y = 0 HOMEWORK SOLVE EACH OF THE FOLLOWING SYSTEMS USING ELIMINATION OR LINEAR COMBINATIONS. Go to PAGE 610 and do 7, 10, 12, 19, 22, 23, 24 28, 30, and 34.

19

SYSTEMS OF LINEAR EQUATIONS

SYSTEMS OF LINEAR EQUATIONS SYSTEMS OF LINEAR EQUATIONS Sstems of linear equations refer to a set of two or more linear equations used to find the value of the unknown variables. If the set of linear equations consist of two equations

More information

5 Systems of Equations

5 Systems of Equations Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

More information

5.2. Systems of linear equations and their solution sets

5.2. Systems of linear equations and their solution sets 5.2. Systems of linear equations and their solution sets Solution sets of systems of equations as intersections of sets Any collection of two or more equations is called a system of equations. The solution

More information

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1

Lesson 9: Graphing Standard Form Equations Lesson 2 of 2. Example 1 Lesson 9: Graphing Standard Form Equations Lesson 2 of 2 Method 2: Rewriting the equation in slope intercept form Use the same strategies that were used for solving equations: 1. 2. Your goal is to solve

More information

Slope-Intercept Equation. Example

Slope-Intercept Equation. Example 1.4 Equations of Lines and Modeling Find the slope and the y intercept of a line given the equation y = mx + b, or f(x) = mx + b. Graph a linear equation using the slope and the y-intercept. Determine

More information

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown.

Warm Up. Write an equation given the slope and y-intercept. Write an equation of the line shown. Warm Up Write an equation given the slope and y-intercept Write an equation of the line shown. EXAMPLE 1 Write an equation given the slope and y-intercept From the graph, you can see that the slope is

More information

6-3 Solving Systems by Elimination

6-3 Solving Systems by Elimination Warm Up Simplify each expression. 1. 2y 4x 2(4y 2x) 2. 5(x y) + 2x + 5y Write the least common multiple. 3. 3 and 6 4. 4 and 10 5. 6 and 8 Objectives Solve systems of linear equations in two variables

More information

2.3 Writing Equations of Lines

2.3 Writing Equations of Lines . Writing Equations of Lines In this section ou will learn to use point-slope form to write an equation of a line use slope-intercept form to write an equation of a line graph linear equations using the

More information

Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library LINEAR EQUATIONS. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library LINEAR EQUATIONS Use this sheet to help you: Solve linear equations containing one unknown Recognize a linear function, and identify its slope and intercept parameters Recognize

More information

Math 215 HW #1 Solutions

Math 215 HW #1 Solutions Math 25 HW # Solutions. Problem.2.3. Describe the intersection of the three planes u+v+w+z = 6 and u+w+z = 4 and u + w = 2 (all in four-dimensional space). Is it a line or a point or an empty set? What

More information

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

More information

Section 3.4 The Slope Intercept Form: y = mx + b

Section 3.4 The Slope Intercept Form: y = mx + b Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept Reminding! m = y x = y 2 y 1 x 2 x 1 Slope of a horizontal line is 0 Slope of a vertical line is Undefined Graph a linear

More information

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0

LINEAR FUNCTIONS. Form Equation Note Standard Ax + By = C A and B are not 0. A > 0 LINEAR FUNCTIONS As previousl described, a linear equation can be defined as an equation in which the highest eponent of the equation variable is one. A linear function is a function of the form f ( )

More information

EdExcel Decision Mathematics 1

EdExcel Decision Mathematics 1 EdExcel Decision Mathematics 1 Linear Programming Section 1: Formulating and solving graphically Notes and Examples These notes contain subsections on: Formulating LP problems Solving LP problems Minimisation

More information

Section 3.2. Graphing linear equations

Section 3.2. Graphing linear equations Section 3.2 Graphing linear equations Learning objectives Graph a linear equation by finding and plotting ordered pair solutions Graph a linear equation and use the equation to make predictions Vocabulary:

More information

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6 Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

Lines and Linear Equations. Slopes

Lines and Linear Equations. Slopes Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level A0 of challenge: D A0 Mathematical goals Starting points Materials required Time needed Connecting perpendicular lines To help learners to: identify perpendicular gradients; identify, from their

More information

Chapter 9. Systems of Linear Equations

Chapter 9. Systems of Linear Equations Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables

More information

Solving Systems of Linear Equations Graphing

Solving Systems of Linear Equations Graphing Solving Systems of Linear Equations Graphing Outcome (learning objective) Students will accurately solve a system of equations by graphing. Student/Class Goal Students thinking about continuing their academic

More information

Solving Systems of Two Equations Algebraically

Solving Systems of Two Equations Algebraically 8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

More information

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

More information

Where Do We Meet? Students will represent and analyze algebraically a wide variety of problem solving situations.

Where Do We Meet? Students will represent and analyze algebraically a wide variety of problem solving situations. Beth Yancey MAED 591 Where Do We Meet? Introduction: This lesson covers objectives in the algebra and geometry strands of the New York State standards for Algebra I. The students will use the graphs of

More information

Voki Lesson Plan. 1. Solve systems of linear and quadratic equations using the graphing calculator

Voki Lesson Plan. 1. Solve systems of linear and quadratic equations using the graphing calculator Voki Lesson Plan Class Title: Integrated Algebra Lesson Title: Solving Linear/Quadratic Systems Grade Level: Grade 9 Author: Steven Viola Objectives: Students will be able to 1. Solve systems of linear

More information

10.1 Systems of Linear Equations: Substitution and Elimination

10.1 Systems of Linear Equations: Substitution and Elimination 726 CHAPTER 10 Systems of Equations and Inequalities 10.1 Systems of Linear Equations: Sustitution and Elimination PREPARING FOR THIS SECTION Before getting started, review the following: Linear Equations

More information

Simple Regression Theory I 2010 Samuel L. Baker

Simple Regression Theory I 2010 Samuel L. Baker SIMPLE REGRESSION THEORY I 1 Simple Regression Theory I 2010 Samuel L. Baker Regression analysis lets you use data to explain and predict. A simple regression line drawn through data points In Assignment

More information

Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form

Overview. Observations. Activities. Chapter 3: Linear Functions Linear Functions: Slope-Intercept Form Name Date Linear Functions: Slope-Intercept Form Student Worksheet Overview The Overview introduces the topics covered in Observations and Activities. Scroll through the Overview using " (! to review,

More information

Systems of Linear Equations

Systems of Linear Equations DETAILED SOLUTIONS AND CONCEPTS - SYSTEMS OF LINEAR EQUATIONS Prepared by Ingrid Stewart, Ph.D., College of Southern Nevada Please Send Questions and Comments to ingrid.stewart@csn.edu. Thank you! PLEASE

More information

Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture

Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture Economics 101 Homework #1 Fall 2014 Due 09/18/2014 in lecture Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the homework

More information

Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

More information

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

More information

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b.

What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of y = mx + b. PRIMARY CONTENT MODULE Algebra - Linear Equations & Inequalities T-37/H-37 What does the number m in y = mx + b measure? To find out, suppose (x 1, y 1 ) and (x 2, y 2 ) are two points on the graph of

More information

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

More information

Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent.

Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. Use the graph to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 2x 1 y = 2x + 3 The lines y = 2x 1 and y = 2x + 3 intersect at exactly one point

More information

Introduction to Diophantine Equations

Introduction to Diophantine Equations Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

More information

2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style

2.7. The straight line. Introduction. Prerequisites. Learning Outcomes. Learning Style The straight line 2.7 Introduction Probably the most important function and graph that you will use are those associated with the straight line. A large number of relationships between engineering variables

More information

Section 1.1 Linear Equations: Slope and Equations of Lines

Section 1.1 Linear Equations: Slope and Equations of Lines Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of

More information

Lesson 22: Solution Sets to Simultaneous Equations

Lesson 22: Solution Sets to Simultaneous Equations Student Outcomes Students identify solutions to simultaneous equations or inequalities; they solve systems of linear equations and inequalities either algebraically or graphically. Classwork Opening Exercise

More information

Graphing Linear Equations in Two Variables

Graphing Linear Equations in Two Variables Math 123 Section 3.2 - Graphing Linear Equations Using Intercepts - Page 1 Graphing Linear Equations in Two Variables I. Graphing Lines A. The graph of a line is just the set of solution points of the

More information

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system

2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system 1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables

More information

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0.

Basic Terminology for Systems of Equations in a Nutshell. E. L. Lady. 3x 1 7x 2 +4x 3 =0 5x 1 +8x 2 12x 3 =0. Basic Terminology for Systems of Equations in a Nutshell E L Lady A system of linear equations is something like the following: x 7x +4x =0 5x +8x x = Note that the number of equations is not required

More information

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)}

Linear Equations. Find the domain and the range of the following set. {(4,5), (7,8), (-1,3), (3,3), (2,-3)} Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in

More information

Alex and Morgan were asked to graph the equation y = 2x + 1

Alex and Morgan were asked to graph the equation y = 2x + 1 Which is better? Ale and Morgan were asked to graph the equation = 2 + 1 Ale s make a table of values wa Morgan s use the slope and -intercept wa First, I made a table. I chose some -values, then plugged

More information

x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m =

x x y y Then, my slope is =. Notice, if we use the slope formula, we ll get the same thing: m = Slope and Lines The slope of a line is a ratio that measures the incline of the line. As a result, the smaller the incline, the closer the slope is to zero and the steeper the incline, the farther the

More information

3.1 Solving Systems Using Tables and Graphs

3.1 Solving Systems Using Tables and Graphs Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system

More information

No Solution Equations Let s look at the following equation: 2 +3=2 +7

No Solution Equations Let s look at the following equation: 2 +3=2 +7 5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

Equations and Inequalities

Equations and Inequalities Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

More information

c sigma & CEMTL

c sigma & CEMTL c sigma & CEMTL Foreword The Regional Centre for Excellence in Mathematics Teaching and Learning (CEMTL) is collaboration between the Shannon Consortium Partners: University of Limerick, Institute of Technology,

More information

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year.

Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students

More information

The Graphical Method: An Example

The Graphical Method: An Example The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

More information

2-4 Writing Linear Equations. Write an equation in slope-intercept form for the line described. 2. passes through ( 2, 3) and (0, 1) SOLUTION:

2-4 Writing Linear Equations. Write an equation in slope-intercept form for the line described. 2. passes through ( 2, 3) and (0, 1) SOLUTION: Write an equation in slope-intercept form for the line described 2 passes through ( 2, 3) and (0, 1) Substitute m = 1 and in the point slope form 4 passes through ( 8, 2); Substitute m = and (x, y) = (

More information

Multiplying Polynomials 5

Multiplying Polynomials 5 Name: Date: Start Time : End Time : Multiplying Polynomials 5 (WS#A10436) Polynomials are expressions that consist of two or more monomials. Polynomials can be multiplied together using the distributive

More information

Determine If An Equation Represents a Function

Determine If An Equation Represents a Function Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

More information

7.3 Solving Systems by Elimination

7.3 Solving Systems by Elimination 7. Solving Sstems b Elimination In the last section we saw the Substitution Method. It turns out there is another method for solving a sstem of linear equations that is also ver good. First, we will need

More information

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES

GRAPHING LINEAR EQUATIONS IN TWO VARIABLES GRAPHING LINEAR EQUATIONS IN TWO VARIABLES The graphs of linear equations in two variables are straight lines. Linear equations may be written in several forms: Slope-Intercept Form: y = mx+ b In an equation

More information

Graphing Equations. with Color Activity

Graphing Equations. with Color Activity Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

Equations of Lines Derivations

Equations of Lines Derivations Equations of Lines Derivations If you know how slope is defined mathematically, then deriving equations of lines is relatively simple. We will start off with the equation for slope, normally designated

More information

I. Model Problems. II. Practice III. Challenge Problems VI. Answer Key

I. Model Problems. II. Practice III. Challenge Problems VI. Answer Key www.mathworksheetsgo.com On Twitter: twitter.com/mathprintables I. Model Problems. II. Practice III. Challenge Problems VI. Answer Key Web Resources Equations of Lines www.mathwarehouse.com/algebra/linear_equation/equation-of-a-line-formula.php

More information

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities

Section P.9 Notes Page 1 P.9 Linear Inequalities and Absolute Value Inequalities Section P.9 Notes Page P.9 Linear Inequalities and Absolute Value Inequalities Sometimes the answer to certain math problems is not just a single answer. Sometimes a range of answers might be the answer.

More information

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

More information

Solving Equations Involving Parallel and Perpendicular Lines Examples

Solving Equations Involving Parallel and Perpendicular Lines Examples Solving Equations Involving Parallel and Perpendicular Lines Examples. The graphs of y = x, y = x, and y = x + are lines that have the same slope. They are parallel lines. Definition of Parallel Lines

More information

WARM UP EXERCSE. 1-3 Linear Functions & Straight lines

WARM UP EXERCSE. 1-3 Linear Functions & Straight lines WARM UP EXERCSE A company makes and sells inline skates. The price-demand function is p (x) = 190 0.013(x 10) 2. Describe how the graph of function p can be obtained from one of the library functions.

More information

1 8 solve quadratic equations by using the quadratic formula and the discriminate September with 16, 2016 notes.noteb

1 8 solve quadratic equations by using the quadratic formula and the discriminate September with 16, 2016 notes.noteb WARM UP 1. Write 15x 2 + 6x = 14x 2 12 in standard form. ANSWER x 2 + 6x +12 = 0 2. Evaluate b 2 4ac when a = 3, b = 6, and c = 5. ANSWER 24 3. A student is solving an equation by completing the square.

More information

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

More information

1 Functions, Graphs and Limits

1 Functions, Graphs and Limits 1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its

More information

Slope-Intercept Form of a Linear Equation Examples

Slope-Intercept Form of a Linear Equation Examples Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation

More information

Graphing Linear Equations

Graphing Linear Equations Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope

More information

Pre-AP Algebra 2 Lesson 1-7 Graphing Absolute Value Functions

Pre-AP Algebra 2 Lesson 1-7 Graphing Absolute Value Functions Lesson 1-7 Graphing Absolute Value Functions Name Objectives: In this activity, students will relate the piecewise function to the graph of the absolute value function and continue their development of

More information

2.5 Zeros of a Polynomial Functions

2.5 Zeros of a Polynomial Functions .5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and

More information

Basic Understandings. Recipes for Functions Guess My Rule!

Basic Understandings. Recipes for Functions Guess My Rule! Activity: TEKS: Recipes for Functions Guess My Rule! (a). (3) Function concepts. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students

More information

Linear Programming. Solving LP Models Using MS Excel, 18

Linear Programming. Solving LP Models Using MS Excel, 18 SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

More information

Question 2: How will changes in the objective function s coefficients change the optimal solution?

Question 2: How will changes in the objective function s coefficients change the optimal solution? Question 2: How will changes in the objective function s coefficients change the optimal solution? In the previous question, we examined how changing the constants in the constraints changed the optimal

More information

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY

Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS 3.2 TERMINOLOGY Chapter 3 LINEAR PROGRAMMING GRAPHICAL SOLUTION 3.1 SOLUTION METHODS Once the problem is formulated by setting appropriate objective function and constraints, the next step is to solve it. Solving LPP

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT1X Intermediate Algebra Workshop I Quadratic Functions LEARNING CENTER Overview Workshop I Quadratic Functions General Form Domain and Range Some of the effects of the leading coefficient a The vertex

More information

2013 MBA Jump Start Program

2013 MBA Jump Start Program 2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

More information

21-114: Calculus for Architecture Homework #1 Solutions

21-114: Calculus for Architecture Homework #1 Solutions 21-114: Calculus for Architecture Homework #1 Solutions November 9, 2004 Mike Picollelli 1.1 #26. Find the domain of g(u) = u + 4 u. Solution: We solve this by considering the terms in the sum separately:

More information

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points?

GRAPHING (2 weeks) Main Underlying Questions: 1. How do you graph points? GRAPHING (2 weeks) The Rectangular Coordinate System 1. Plot ordered pairs of numbers on the rectangular coordinate system 2. Graph paired data to create a scatter diagram 1. How do you graph points? 2.

More information

Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c.

Name: Class: Date: Does the equation represent a direct variation? If so, find the constant of variation. c. yes; k = 5 3. c. Name: Class: Date: Chapter 5 Test Multiple Choice Identify the choice that best completes the statement or answers the question. What is the slope of the line that passes through the pair of points? 1.

More information

Practice Test - Chapter 4. y = 2x 3. The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept.

Practice Test - Chapter 4. y = 2x 3. The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept. y = 2x 3. The slope-intercept form of a line is y = mx + b, where m is the slope, and b is the y-intercept. Plot the y-intercept (0, 3). The slope is. From (0, 3), move up 2 units and right 1 unit. Plot

More information

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , )

Algebra I Pacing Guide Days Units Notes 9 Chapter 1 ( , ) Algebra I Pacing Guide Days Units Notes 9 Chapter 1 (1.1-1.4, 1.6-1.7) Expressions, Equations and Functions Differentiate between and write expressions, equations and inequalities as well as applying order

More information

Grade 8 Mathematics Item Specification C1 TD Task Model 3

Grade 8 Mathematics Item Specification C1 TD Task Model 3 Task Model 3 Equation/Numeric DOK Level 1 algebraically, example, have no solution because 6. 3. The student estimates solutions by graphing systems of two linear equations in two variables. Prompt Features:

More information

with "a", "b" and "c" representing real numbers, and "a" is not equal to zero.

with a, b and c representing real numbers, and a is not equal to zero. 3.1 SOLVING QUADRATIC EQUATIONS: * A QUADRATIC is a polynomial whose highest exponent is. * The "standard form" of a quadratic equation is: ax + bx + c = 0 with "a", "b" and "c" representing real numbers,

More information

PPS TI-83 Activities Algebra 1-2 Teacher Notes

PPS TI-83 Activities Algebra 1-2 Teacher Notes PPS TI-83 Activities Algebra 1-2 Teacher Notes It is an expectation in Portland Public Schools that all teachers are using the TI-83 graphing calculator in meaningful ways in Algebra 1-2. This does not

More information

Write the Equation of the Line Review

Write the Equation of the Line Review Connecting Algebra 1 to Advanced Placement* Mathematics A Resource and Strategy Guide Objective: Students will be assessed on their ability to write the equation of a line in multiple methods. Connections

More information

Corinne: I m thinking of a number between 220 and 20. What s my number? Benjamin: Is it 25?

Corinne: I m thinking of a number between 220 and 20. What s my number? Benjamin: Is it 25? Walk the Line Adding Integers, Part I Learning Goals In this lesson, you will: Model the addition of integers on a number line. Develop a rule for adding integers. Corinne: I m thinking of a number between

More information

Lecture 5: Correlation and Linear Regression

Lecture 5: Correlation and Linear Regression Lecture 5: Correlation and Linear Regression 3.5. (Pearson) correlation coefficient The correlation coefficient measures the strength of the linear relationship between two variables. The correlation is

More information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form Concepts # and # Sect. - The Slope-Intercept Form Slope-Intercept Form of a line Recall the following definition from the beginning of the chapter: Let a, b, and c be real numbers where a and b are not

More information

Section 2.3. Learning Objectives. Graphing Quadratic Functions

Section 2.3. Learning Objectives. Graphing Quadratic Functions Section 2.3 Quadratic Functions Learning Objectives Quadratic function, equations, and inequities Properties of quadratic function and their graphs Applications More general functions Graphing Quadratic

More information

MATH 111: EXAM 02 SOLUTIONS

MATH 111: EXAM 02 SOLUTIONS MATH 111: EXAM 02 SOLUTIONS BLAKE FARMAN UNIVERSITY OF SOUTH CAROLINA Answer the questions in the spaces provided on the question sheets and turn them in at the end of the class period Unless otherwise

More information

Objectives A c t i v i t y 1 How Many Drivers? Investigating the Slope- Intercept Form of a Line Introduction

Objectives A c t i v i t y 1 How Many Drivers? Investigating the Slope- Intercept Form of a Line Introduction Objectives Activity 1 Recognize the slope-intercept form of a linear equation Investigate the effects of changes in A and B on a linear equation Use the guess-and-check strategy to manipulate the values

More information

Linear Equations Review

Linear Equations Review Linear Equations Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The y-intercept of the line y = 4x 7 is a. 7 c. 4 b. 4 d. 7 2. What is the y-intercept

More information

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2).

Chapter 2 Section 4: Equations of Lines. 4.* Find the equation of the line with slope 4 3, and passing through the point (0,2). Chapter Section : Equations of Lines Answers to Problems For problems -, put our answers into slope intercept form..* Find the equation of the line with slope, and passing through the point (,0).. Find

More information

with functions, expressions and equations which follow in units 3 and 4.

with functions, expressions and equations which follow in units 3 and 4. Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model

More information

Question 2: How do you solve a linear programming problem with a graph?

Question 2: How do you solve a linear programming problem with a graph? Question 2: How do you solve a linear programming problem with a graph? Now that we have several linear programming problems, let s look at how we can solve them using the graph of the system of inequalities.

More information

Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin.

Positive numbers move to the right or up relative to the origin. Negative numbers move to the left or down relative to the origin. 1. Introduction To describe position we need a fixed reference (start) point and a way to measure direction and distance. In Mathematics we use Cartesian coordinates, named after the Mathematician and

More information

price quantity q The Supply Function price quantity q

price quantity q The Supply Function price quantity q Shown below is another demand function for price of a pizza p as a function of the quantity of pizzas sold per week. This function models the behavior of consumers with respect to price and quantity. 3

More information

You might be surprised to know that the word T-shirt wasn t really used until

You might be surprised to know that the word T-shirt wasn t really used until Hot Shirts Using Tables, Graphs, and Equations, Part 2 Learning Goals In this lesson, you will: Use different methods to represent a problem situation. Estimate values of expressions that involve decimals.

More information