# Solving Equations Involving Parallel and Perpendicular Lines Examples

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 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 In a plane, lines with the same slope are parallel lines. Also, all vertical lines are parallel.. Example Find the slope of a line parallel to the line whose equation is y x =. Parallel lines have the same slope. Find the slope of the line whose equation is y x =. To do so, write the equation in slope-intercept form (). y x = y = x + y = x + The slope of any line parallel to the given line is.. Example Find the slope of a line parallel to the line whose equation is y x = Parallel lines have the same slope. Find the slope of the line whose equation is y x =. To do so, write the equation in slope-intercept form (). y = x The slope of any line parallel to the given line is.

2 . Example Find an equation of the line that passes through (, 6) and is parallel to the line whose equation is y = x +. The slope is (notice that y = x + is in slope-intercept form. Use (, 6) and the slope to find the y-intercept. 6 = ( )() + b 6 = 8 +b 0 = b Substitution Property 0 An equation of the line is y = x +. Thought Provoker What is the relationship between the x- and y-intercepts of parallel lines? If the intercepts are nonzero, the ratio of the x- and y-intercepts of a given line is equal to the ratio of the x- and y-intercepts of any line parallel to the given line. 6. Example Find an equation of the line that passes through (, ) and is parallel to y x = Rewrite y x = into slope-intercept form y = x + The slope of all parallel lines must be. Find b by substituting into slope-intercept form = ( ) + b b = 0 Therefore y = x + 0 must be the equation of a line passing through (, ) and parallel to y x =.

3 7. Example Find an equation of the line that passes through (, ) and is parallel to x + y = 6. Rewrite x + y = 6 into slope-intercept form y = The slope of all parallel lines must be. 6 x + Find b by substituting into slope-intercept form = ( ) + b b = Therefore y = x and parallel to x + y = must be the equation of a line passing through (, ) 8. The graphs of y = x + and y = x + 6 are lines that are perpendicular. Notice how their slopes are related: ( )( ) = The product of their slopes is 9. Definition of Perpendicular Lines In a plane, two nonvertical lines are perpendicular if and only if the product of their slopes is. Any vertical line is perpendicular to any horizontal line. 0. Here is a way to show that the slopes of any two nonvertical perpendicular lines in a plane have a product of. Consider a line that is neither vertical nor horizontal, with slope s r (see graph). Now consider rotating the line Notice that the slope r of the new line (see graph) is. The product of s r r the slopes of the two lines is ( )( ) =. s s

4 . Example Find the slope of a line perpendicular to the line whose equation is y x =. y x = y = x + Write equation in slope-intercept form. The slope of the given line is. (m) = m = Let m stand for the slope of the perpendicular line. The slope of any line perpendicular to the given line is. Example Find the slope of a line perpendicular to the line whose equation is x 7y = 6. x 7y = 6 6 y = x 7 7 Write equation in slope-intercept form. The slope is 7 (m) = 7 7 m = Let m stand for the slope of the perpendicular line. The slope of any line perpendicular to the given line is 7

5 . Example Find an equation of the line that passes through (, 6) and is perpendicular to the line whose equation is y = x +. The slope of the given line is. (m) = m = Use (, 6) and the slope 6 = ( )() + b = b Let m stand for the slope of the perpendicular line. to find the y-intercept. The y-intercept is. An equation of the line is y = x + This could be written x + y =. Example Find an equation of a line passing through (, ) and is perpendicular to x + y = 6. y = x + 6 The slope of the given line is. (m) = Let m stand for the slope of the perpendicular line. m = Use (, ) and the slope to find the y-intercept. = ()( ) + b The y-intercept is 0. 0 = b An equation of the line is y = x This could be written x y = 0

6 . Example Find an equation of a line that passes through (, ) and is perpendicular x + y =. y = x + The slope of the given line is (m) = m =. Let m stand for the slope of the perpendicular line. Use (, ) and the slope to find the y-intercept. = ( )() + b = b The y-intercept is An equation of the line is y = x. This could be written x y = 6. Thought Provoker Is it possible that two lines represented by 6x y = and x + y = 7 are perpendicular? Rewrite in slope-intercept form. 6x y = y = 6x + y = x + x + y = 7 y = x y = x + Yes. They are perpendicular. 6

7 Name: Date: Class: Solving Equations Involving Parallel and Perpendicular Lines Worksheet Find the slope of a line that is parallel and the slope of a line that is perpendicular to each line whose equation is given.. y = x +. y = x. x y = 8. y = x 8 6. x = y y x = 0 State whether the graphs of the following equations are parallel, perpendicular, or neither. 7. x + y = 8. x + y = x + y = 0 x y = 9. y = x 0. y + x = y = x y x =. x 8y = x 6y = 0. y + x = y + x =. x + y =. x + y = x + y = 7 x y = 7

8 Find an equation of the line that passes through each given point and is parallel to the line with the given equation.. (, ); y = x 6. (, ); y = x (, ); x + y = 8. (0, 0); x y = Find an equation of the line that passes through each given point and is perpendicular to the line with the given equation. 9. (, 0); y = x (, ); x + y = 7. (, 6); x + y =. (0, ); 6x y = Find the value of a for which the graph of the first equation is perpendicular to the graph of the second equation.. y = ax ; y = x. y = ax + ; y x = 7. y = a x 6; x + y = 6 6. y + ax = 8; y = x + 8

9 Name: Date: Class: Solving Equations Involving Parallel and Perpendicular Lines Worksheet Key Find the slope of a line that is parallel and the slope of a line that is perpendicular to each line whose equation is given.. y = x + Parallel slope = ; Perpendicular slope =. y = x y = x + Parallel slope = ; Perpendicular slope =. y = x 8 y = x Parallel slope = ; Perpendicular slope =. 6y x = 0 y = 6 x 6 Parallel slope = ; Perpendicular slope = 6. x y = y = x Parallel slope = ; Perpendicular slope = x = y y = x Parallel slope = ; Perpendicular slope = State whether the graphs of the following equations are parallel, perpendicular, or neither. y = x + Both have same slope. 7. x + y = y = x 0 Lines are parallel. x + y = 0 9

10 8. x + y = x y = y = x + y = x The product of the slopes is ()( ) = Lines are perpendicular. 9. y = x y = x Lines have same slope. y = x y = x Lines are parallel. 0. y + x = y x = y = x + y = x + The product of the slopes is ( )( ) = Lines are perpendicular.. x 8y = y = x 8 8 Neither x 6y = 0 y = x. y + x = y + x = y = x + y = x + Neither. x + y = x + y = 7 9 y = x y = x + Lines have same slope. Lines are parallel.. x + y = x y = y = x + 6 y = x The product of the slopes is ( )( ) = Lines are perpendicular. 0

11 Find an equation of the line that passes through each given point and is parallel to the line with the given equation.. (, ); y = x Use m = and (, ) to find b. = () + b b = 6 y = x 6 6. (, ); y = x + 6 Use m = and (, ) to find b. = () + b b = 0 y = x 7. (, ); x + y = x + y = y = x + Use m = and (, ); to find b. = ( ) + b 7 7 b = y = x + 8. (0, 0); x y = x y = y = x Use m = and (0, 0); to find b. 0 = (0) + b b = 0 y = x

12 Find an equation of the line that passes through each given point and is perpendicular to the line with the given equation. 9. (, 0); y = x + 7 m = For a perpendicular line use m = and (, 0) 0 = ( ) + b b = y = x + 0. (, ); x + y = 7 x + y = 7 y = m = 7 x + For a perpendicular line use m = and (, ) = () + b b = y = x +. (0, ); 6x y = 6x y = y = x m = For a perpendicular line use m = = b = (0) + b y = x and (0, )

13 . (, 6); x + y = x + y = y = x + m = For a perpendicular line use m = and (, 6) 6 = () + b b = y = x Find the value of a for which the graph of the first equation is perpendicular to the graph of the second equation.. y = ax ; y = x Rewrite y = x in slope-intercept form. y = x Compare slopes with y = ax. If a has a perpendicular slope then (a) = a =. y = ax + ; y x = 7 Rewrite y x = 7 in slope-intercept form. y = x + 7 Compare slopes with y = ax +. If a has a perpendicular slope then (a) = a =

14 . y = a x 6; x + y = 6 Rewrite x + y = 6 in slope-intercept form. a y = x + 6 Compare slopes with y = x 6. a If a has a perpendicular slope then ( ) = a = 6. y + ax = 8; y = x + Rewrite y + ax = 8 in slope-intercept form. a 8 y = x + Compare slopes with y = x +. a If a has a perpendicular slope then ( ) = a =

15 Student Name: Date: Solving Equations Involving Parallel and Perpendicular Lines Checklist. On questions thru 6, did the student find the slope of a parallel line? a. Yes (0 points) b. out of 6 ( points) c. out of 6 (0 points) d. out of 6 ( points) e. out of 6 (0 points) f. out of 6 ( points). On questions thru 6, did the student find the slope of a perpendicular line? a. Yes (0 points) b. out of 6 ( points) c. out of 6 (0 points) d. out of 6 ( points) e. out of 6 (0 points) f. out of 6 ( points). On questions 7 thru, did the student correctly determine the type of graphs? a. Yes (0 points) b. 7 out of 8 ( points) c. 6 out of 8 (0 points) d. out of 8 ( points) e. out of 8 (0 points) f. out of 8 ( points) g. out of 8 (0 points) h. out of 8 ( points). On questions thru 8, did the student find the equation of a parallel line correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points). On questions 9 thru, did the student find the equation of a perpendicular line correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points) 6. On questions thru 6, did the student solve for a correctly? a. Yes (0 points) b. out of ( points) c. out of (0 points) d. out of ( points) Total Number of Points

16 Name: Date: Class: NOTE: The sole purpose of this checklist is to aide the teacher in identifying students that need remediation. It is suggested that teacher s devise their own point range for determining grades. In addition, some students need remediation in specific areas. The following checklist provides a means for the teacher to assess which areas need addressing.. Does the student need remediation in content (finding the slope of a parallel and perpendicular line when given a linear equation) for questions thru 6? Yes No. Does the student need remediation in content (identifying parallel and perpendicular lines when given a pair of linear equations) for questions 7 thru? Yes No. Does the student need remediation in content (finding the equation of a parallel line when given a point and a linear equation) for questions thru 8? Yes No. Does the student need remediation in content (finding the equation of a perpendicular line when given a point and a linear equation) for questions 9 thru? Yes No. Does the student need remediation in content (finding the value of the constant term when given a perpendicular line) for questions thru 6? Yes No A B C D F 0 points and above 6 points and above points and above 96 points and above 9 points and below Sample Range of Points. 6

### 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

### 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

### 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

### 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

### Solving Systems of Equations Algebraically Examples

Solving Systems of Equations Algebraically Examples 1. Graphing a system of equations is a good way to determine their solution if the intersection is an integer. However, if the solution is not an integer,

### 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

### 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

### 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

### 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

### 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

### 5 \$75 6 \$90 7 \$105. Name Hour. Review Slope & Equations of Lines. STANDARD FORM: Ax + By = C. 1. What is the slope of a vertical line?

Review Slope & Equations of Lines Name Hour STANDARD FORM: Ax + By = C 1. What is the slope of a vertical line? 2. What is the slope of a horizontal line? 3. Is y = 4 the equation of a horizontal or vertical

### 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

### 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.

### 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

### Writing the Equation of a Line in Slope-Intercept Form

Writing the Equation of a Line in Slope-Intercept Form Slope-Intercept Form y = mx + b Example 1: Give the equation of the line in slope-intercept form a. With y-intercept (0, 2) and slope -9 b. Passing

### 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

1.3 LINEAR EQUATIONS IN TWO VARIABLES Copyright Cengage Learning. All rights reserved. What You Should Learn Use slope to graph linear equations in two variables. Find the slope of a line given two points

### In this section, we ll review plotting points, slope of a line and different forms of an equation of a line.

Math 1313 Section 1.2: Straight Lines In this section, we ll review plotting points, slope of a line and different forms of an equation of a line. Graphing Points and Regions Here s the coordinate plane:

### Pre-Calculus III Linear Functions and Quadratic Functions

Linear Functions.. 1 Finding Slope...1 Slope Intercept 1 Point Slope Form.1 Parallel Lines.. Line Parallel to a Given Line.. Perpendicular Lines. Line Perpendicular to a Given Line 3 Quadratic Equations.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

### 5. Equations of Lines: slope intercept & point slope

5. Equations of Lines: slope intercept & point slope Slope of the line m rise run Slope-Intercept Form m + b m is slope; b is -intercept Point-Slope Form m( + or m( Slope of parallel lines m m (slopes

### Number of Solutions to Simultaneous Equations

Worksheet 3.5 Simultaneous Equations Section 1 Number of Solutions to Simultaneous Equations In maths we are sometimes confronted with two equations in two variables and we want to find out which values

### Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

### 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

### Study Guide and Review - Chapter 4

State whether each sentence is true or false. If false, replace the underlined term to make a true sentence. 1. The y-intercept is the y-coordinate of the point where the graph crosses the y-axis. The

### 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

### Section 2.2 Equations of Lines

Section 2.2 Equations of Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes

### 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

### 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

### 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

### Section 1.4 Notes Page Linear Equations in Two Variables and Linear Functions., x

Section. Notes Page. Linear Equations in Two Variables and Linear Functions Slope Formula The slope formula is used to find the slope between two points ( x, y ) and ( ) x, y. x, y ) The slope is the vertical

### Section 1.10 Lines. The Slope of a Line

Section 1.10 Lines The Slope of a Line EXAMPLE: Find the slope of the line that passes through the points P(2,1) and Q(8,5). = 5 1 8 2 = 4 6 = 2 1 EXAMPLE: Find the slope of the line that passes through

### Math 10 - Unit 7 Final Review - Coordinate Geometry

Class: Date: Math 10 - Unit Final Review - Coordinate Geometry Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Determine the slope of this line segment.

### Tangent line of a circle can be determined once the tangent point or the slope of the line is known.

Worksheet 7: Tangent Line of a Circle Name: Date: Tangent line of a circle can be determined once the tangent point or the slope of the line is known. Straight line: an overview General form : Ax + By

### Pre-AP Algebra 2 Lesson 2-5 Graphing linear inequalities & systems of inequalities

Lesson 2-5 Graphing linear inequalities & systems of inequalities Objectives: The students will be able to - graph linear functions in slope-intercept and standard form, as well as vertical and horizontal

### Example SECTION 13-1. X-AXIS - the horizontal number line. Y-AXIS - the vertical number line ORIGIN - the point where the x-axis and y-axis cross

CHAPTER 13 SECTION 13-1 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants X-AXIS - the horizontal

### Linear Equations and Graphs

2.1-2.4 Linear Equations and Graphs Coordinate Plane Quadrants - The x-axis and y-axis form 4 "areas" known as quadrants. 1. I - The first quadrant has positive x and positive y points. 2. II - The second

### 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

### 5.1: Rate of Change and Slope

5.1: Rate of Change and Slope Rate of Change shows relationship between changing quantities. On a graph, when we compare rise and run, we are talking about steepness of a line (slope). You can use and

### Algebra 1 Chapter 05 Review

Class: Date: Algebra 1 Chapter 05 Review Multiple Choice Identify the choice that best completes the statement or answers the question. Find the slope of the line that passes through the pair of points.

### Chapter 4.1 Parallel Lines and Planes

Chapter 4.1 Parallel Lines and Planes Expand on our definition of parallel lines Introduce the idea of parallel planes. What do we recall about parallel lines? In geometry, we have to be concerned about

### Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope

Math 152 Rodriguez Blitzer 2.4 Linear Functions and Slope I. Linear Functions 1. A linear equation is an equation whose graph is a straight line. 2. A linear equation in standard form: Ax +By=C ex: 4x

### Solving Inequalities Examples

Solving Inequalities Examples 1. Joe and Katie are dancers. Suppose you compare their weights. You can make only one of the following statements. Joe s weight is less than Kate s weight. Joe s weight is

### MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines

MATH 105: Finite Mathematics 1-1: Rectangular Coordinates, Lines Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006 Outline 1 Rectangular Coordinate System 2 Graphing Lines 3 The Equation of

### Portable Assisted Study Sequence ALGEBRA IIA

SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

### How Many Solutions? Resource ID#: Primary Type: Formative Assessment

How Many Solutions? Resource ID#: 59692 Primary Type: Formative Assessment This document was generated on CPALMS - www.cpalms.org Students are asked to determine the number of solutions of each of four

### 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.

### 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,,

### COMPARING LINEAR AND NONLINEAR FUNCTIONS

1 COMPARING LINEAR AND NONLINEAR FUNCTIONS LEARNING MAP INFORMATION STANDARDS 8.F.2 Compare two s, each in a way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example,

### Creating Equations. Set 3: Writing Linear Equations Instruction. Student Activities Overview and Answer Key

Creating Equations Instruction Goal: To provide opportunities for students to develop concepts and skills related to writing linear equations in slope-intercept and standard form given two points and a

### 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

### 4.1 & Linear Equations in Slope-Intercept Form

4.1 & 4.2 - Linear Equations in Slope-Intercept Form Slope-Intercept Form: y = mx + b Ex 1: Write the equation of a line with a slope of -2 and a y-intercept of 5. Ex 2:Write an equation of the line shown

### Lecture 9: Lines. m = y 2 y 1 x 2 x 1

Lecture 9: Lines If we have two distinct points in the Cartesian plane, there is a unique line which passes through the two points. We can construct it by joining the points with a straight edge and extending

### 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

### 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

### Determining Angle Measure with Parallel Lines Examples

Determining Angle Measure with Parallel Lines Examples 1. Using the figure at the right, review with students the following angles: corresponding, alternate interior, alternate exterior and consecutive

### Students will use various media (computer, graphing calculator, paper and pencil) to graph/sketch linear equations.

Title: Lines, Lines, Everywhere!! A discovery/exploration lesson investigating equations of the form y = mx + b to see how the values of b and m affects the graph. Link to Outcomes: Communication/ Cooperation

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

Use the graph shown to determine whether each system is consistent or inconsistent and if it is independent or dependent. y = 3x + 1 y = 3x + 1 The two equations intersect at exactly one point, so they

### 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.

### Unit 5: Coordinate Geometry Practice Test

Unit 5: Coordinate Geometry Practice Test Math 10 Common Name: Block: Please initial this box to indicate you carefully read over your test and checked your work for simple mistakes. What I can do in this

### The Parabola and the Circle

The Parabola and the Circle The following are several terms and definitions to aid in the understanding of parabolas. 1.) Parabola - A parabola is the set of all points (h, k) that are equidistant from

### Algebra 1 Chapter 3 Vocabulary. equivalent - Equations with the same solutions as the original equation are called.

Chapter 3 Vocabulary equivalent - Equations with the same solutions as the original equation are called. formula - An algebraic equation that relates two or more real-life quantities. unit rate - A rate

### A synonym is a word that has the same or almost the same definition of

Slope-Intercept Form Determining the Rate of Change and y-intercept Learning Goals In this lesson, you will: Graph lines using the slope and y-intercept. Calculate the y-intercept of a line when given

### Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan

Lines, Lines, Lines!!! Slope-Intercept Form ~ Lesson Plan I. Topic: Slope-Intercept Form II. III. Goals and Objectives: A. The student will write an equation of a line given information about its graph.

### Midterm 1. Solutions

Stony Brook University Introduction to Calculus Mathematics Department MAT 13, Fall 01 J. Viro October 17th, 01 Midterm 1. Solutions 1 (6pt). Under each picture state whether it is the graph of a function

### What students need to know for... ALGEBRA II

What students need to know for... ALGEBRA II 2015-2016 NAME Students expecting to take Algebra II at Cambridge Rindge & Latin School should demonstrate the ability to... General: o Keep an organized notebook

### Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

### 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

### 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 ( )

### 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

### Section summaries. d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2. 1 + y 2. x1 + x 2

Chapter 2 Graphs Section summaries Section 2.1 The Distance and Midpoint Formulas You need to know the distance formula d = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 and the midpoint formula ( x1 + x 2, y ) 1 + y 2

### The Perpendicular Bisector of a Segment

Check off the steps as you complete them: The Perpendicular Bisector of a Segment Open a Geometry window in Geogebra. Use the segment tool of draw AB. Under the point tool, select Midpoint of Center, and

### HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

### Worksheet A5: Slope Intercept Form

Name Date Worksheet A5: Slope Intercept Form Find the Slope of each line below 1 3 Y - - - - - - - - - - Graph the lines containing the point below, then find their slopes from counting on the graph!.

### 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

### Vocabulary Words and Definitions for Algebra

Name: Period: Vocabulary Words and s for Algebra Absolute Value Additive Inverse Algebraic Expression Ascending Order Associative Property Axis of Symmetry Base Binomial Coefficient Combine Like Terms

### Geometry 1. Unit 3: Perpendicular and Parallel Lines

Geometry 1 Unit 3: Perpendicular and Parallel Lines Geometry 1 Unit 3 3.1 Lines and Angles Lines and Angles Parallel Lines Parallel lines are lines that are coplanar and do not intersect. Some examples

### 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

### Final Graphing Practice #1

Final Graphing Practice #1 Beginning Algebra / Math 100 Fall 2013 506 (Prof. Miller) Student Name/ID: Instructor Note: Assignment: Set up a tutoring appointment with one of the campus tutors or with me.

### Name: Date: Block: Midterm Exam Review Sheet 1

Name: Date: Block: Midterm Exam Review Sheet 1 Chapter 1 1. Write a variable expression for 2. Simplify: 8 3 2 4 seven divided by the sum of x and five 3. Write an algebraic expression for 4. Write an

### Section 1.8 Coordinate Geometry

Section 1.8 Coordinate Geometry The Coordinate Plane Just as points on a line can be identified with real numbers to form the coordinate line, points in a plane can be identified with ordered pairs of

### Algebra I: Strand 2. Linear Functions; Topic 2. Formalizing Slope and y-intercept; Task

1 TASK : DESCRIBE STACKING CUPS Solutions Make a scatter plot of the data. Write a symbolic function rule and a verbal description of the situation. 1. 1 1 1 Number of Cups 0.5 1 6.0 2 8.5 11.0 2.5 2.5

### Design a Brochure Algebra 2 Project Linear Equations and Inequalities

Design a Brochure Algebra 2 Project Linear Equations and Inequalities Purpose: For this project, you will be working as a group of two to design a tri-fold brochure or an alternate display that will summarize

### Algebra. Indiana Standards 1 ST 6 WEEKS

Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

### 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

### MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem. Constant Rate of Change/Slope

MA.8.A.1.2 Interpret the slope and the x- and y-intercepts when graphing a linear equation for a real-world problem Constant Rate of Change/Slope In a Table Relationships that have straight-lined graphs

### Exam 2 Review. 3. How to tell if an equation is linear? An equation is linear if it can be written, through simplification, in the form.

Exam 2 Review Chapter 1 Section1 Do You Know: 1. What does it mean to solve an equation? To solve an equation is to find the solution set, that is, to find the set of all elements in the domain of the

### ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

### 2.1 Equations of Lines

Section 2.1 Equations of Lines 1 2.1 Equations of Lines The Slope-Intercept Form Recall the formula for the slope of a line. Let s assume that the dependent variable is and the independent variable is

### Unit III Practice Questions

Unit III Practice Questions 1. Write the ordered pair corresponding to each point plotted below. A F B E D C 2. Determine if the ordered pair ( 1, 2) is a solution of 2x + y = 4. Explain how you know.

### 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:

### Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

### -2- Reason: This is harder. I'll give an argument in an Addendum to this handout.

LINES Slope The slope of a nonvertical line in a coordinate plane is defined as follows: Let P 1 (x 1, y 1 ) and P 2 (x 2, y 2 ) be any two points on the line. Then slope of the line = y 2 y 1 change in

### Slope-Intercept Quiz. Name: Class: Date: 1. Graph the line with the slope 1 and y-intercept 3. a. c. b. d.

Name: Class: Date: ID: A Slope-Intercept Quiz 1. Graph the line with the slope 1 and y-intercept 3. a. c. b. d.. Write the equation that describes the line with slope = and y-intercept = 3 a. x + y = 3

### Grade 10 Applied Math Unit 3: Linear Equations Lesson: Date: Completed Level Lesson 1: Coordinates Review Lesson 2: Constructing Tables and Sketching

Grade 10 Applied Math Unit 3: Linear Equations Lesson: Date: Completed Level Lesson 1: Coordinates Review Lesson 2: Constructing Tables and Sketching Graphs Lesson 3: Interpreting Lines Lesson 4: Write

### 2.1 Algebraic Expressions and Combining like Terms

2.1 Algebraic Expressions and Combining like Terms Evaluate the following algebraic expressions for the given values of the variables. 3 3 3 Simplify the following algebraic expressions by combining like

### Lesson 8.3 Exercises, pages

Lesson 8. Eercises, pages 57 5 A. For each function, write the equation of the corresponding reciprocal function. a) = 5 - b) = 5 c) = - d) =. Sketch broken lines to represent the vertical and horizontal

### Lecture 8 : Coordinate Geometry. The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 20

Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding