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

Size: px
Start display at page:

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

Transcription

1 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 equations. Graphs of linear functions. Simplifying Algebraic Expressions Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions. 3x + 4 5x 2y + 6 x 2 + 5x 4 x 4 16 x 2 + 5xy + 6y 2 The terms of an algebraic expression are those quantities that are separated by addition. Example: List the terms of the algebraic expression 4x 2 3x + 6 Terms are identified by the addition operation. Write the expression using addition.. Solution 4x 2 3x + 6 Write the expression using addition. 4x 2 + ( 3x)+ 6 Subtraction can be written in terms of addition: a b = a + ( b) 4x 2, 3x and 6 Identify the terms in the expression.

2 Page 2 of 13 The coefficient of a term is its numerical factor. Example: Determine the coefficients of the following terms. 5x 2, 2xy 3, x2 y 3 4, a2 b. Factors are quantities that are multiplied together. Think of each term as a product of a number (the coefficient) and variables. Solution 5x 2 The coefficient is 5. 2xy 2 The coefficient is 2. x 2 y 3 4 = 1 4 x2 y 3 The coefficient is 1 4. a 2 b = 1 a 2 b The coefficient is 1. Like terms (also known as common terms or similar terms) are terms that have the same variable factors. Example: Determine the like terms in the following list. 5x 2, 3y 2,8xy,2x 2 y,2xy,7x 2 y,10x,and9y 2 Compare the variable factors in two terms. For example, when considering the term 7x 2 y, the variable factors are x 2 y. (Note that 7 is the coefficient.) Solution 3y 2 and 9y 2 are like terms. The variable factors are y 2 in both terms. 8xy and 2xy are like terms. The variable factors are xy in both terms. 2x 2 y and 7x 2 y are like terms. The variable factors are x 2 y in both terms. Note that there are two terms left over: the same ( x 2 and x respectively). 5x 2 and 10x. These two terms are not like terms because their variable factors are not

3 Page 3 of 13 The properties of the real numbers are used when working with algebraic expression. Consider the following examples. Properties of Real Numbers Property Example Commutative Property of Addition a + b = b + a 14x + 5x 2 = 5x 2 +14x Commutative Property of Multiplication a b = b a x 3 7 = 7x 3 Associative Property of Addition ( a + b)+ c = a + ( b + c) ( 3x 2 + 4x)+ 7x = 3x 2 + 4x + 7x Associative Property of Multiplication ( a b) c = a ( b c) 54x ( )= ( 5 4)x Distributive Property ab+ ( c)= a b + a c 5( x + 2)= 5x Can also be written as: b + c ( )a = b a + c a ( 4 + y)x = 4 x + y x ( ) The distributive property allows us to add or subtract like terms. The adding (or subtracting) of s like terms is known as combining like terms. Example: Combine the like terms 3x 2 + 5x 2 The two terms being added are like terms. Their variable factors are both x 2. Therefore it is possible to combine them using the distributive property. Solution 3x 2 + 5x 2 2 ( 3+ 5) x Use the distributive property. 8x 2

4 Page 4 of 13 Example: Combine the like terms: 3x 2 + 4x 2x 2 + 3x 4 There are five terms in this expression. Identify the like terms. Solution 3x 2 + 4x 2x 2 + 3x 4 3x 2 and 2x 2 are like terms as are 4x and 3x. 3x 2 2x 2 + 4x + 3x 4 Use the commutative property of addition to bring like terms together. ( 3 2)x + ( 4 + 3)x 4 Use of the distributive property to combine like terms. x 2 + 7x 4 To simplify an algebraic expression with parenthesis use the distributive property, then combine like terms. Example: Simplify 2( 5x 1)+14x Before any terms can be combined in this expression the distributive property must be used in order to remove the parenthesis. Solution 25x ( 1)+14x 10x 2 +14x 24x 2 Use of the distributive property. Example: Simplify 5( x 2 + 2x) x( 2x 4) There are two sets of parenthesis so the distributive property will be used twice. Solution 5( x 2 + 2x) x( 2x 4) 5x 2 +10x 2x 2 + 4x 5x 2 2x 2 +10x + 4x 3x 2 +14x Use of the distributive property. Use of the commutative property of addition to bring like terms together.

5 Page 5 of 13 Linear Expressions The degree of a term is the sum of the exponents on the variable factors. Example: Determine the degree of 5x 2 y. Identify the exponents on the variable factor, to determine the degree of the terms. Solution 5x 2 y x 2 and y Identify the variable factors. 2 and 1 Identify the exponents on the variable factors. 2 +1or 3 Add the exponents to determine the degree of the term. Note: The degree of a constant (e.g. 6) is 0, since Example: Determine the degree of the constant = 6x. Since the degree of a term is based on the variable factor, rewrite the term using a variable (recall that x 0 = 1). Solution 8 8x 0 Rewrite 8 with a variable factor. 0 The degree of the term.

6 Page 6 of 13 A linear expression is a sum (or difference) of terms in which the highest degree of any term is 1. Example: Determine if the expression 3x 4 is linear. To determine whether or not the expression is linear we must determine the degree of each term. Solution 3x and 4 List the two terms of the expression. 1 and 0 Identify the degree of 3x and 4 respectively. The expression is linear. Since the highest degree is 1, the expression is linear. Example: Determine whether the expression 3( x 4) 2( x 6) is linear. Since there are parenthesis, before determining the degree of each term, simplify the expression. Solution 3( x 5) 2( x 6) 3x 15 2x +12 Use of distributive property. x 3 x and 3 List the terms. 1 and 0 Identify the degree of x and 3 respectively. The expression is linear. Since the highest degree is 1, the expression is linear. Example: Determine if the expression 3x 2 4x + 7 is linear. To determine whether or not the expression is linear, determine the degree of each term. Solution 3x 2, 4x,and,7 List the three terms of the expression. 2, 1 and 0 Identify the degree of 3x 2, 4x, and, 7 respectively. The expression in not linear. The highest degree is greater than 1, so the expression is not linear.

7 Page 7 of 13 Solving Linear Equations A linear equation (in one variable x) is an equation that can be written in the form ax + b = 0. Two properties are used to solve linear equations. Addition Property of Equality: The same number (or algebraic expression) can be added to both sides of an equation without changing the equation s solution set. Multiplication Property of Equality: The same nonzero number may multiply both sides of an equation without changing the equation s solution set. Example: Solve the equation. x 4 = 7 This is a linear equation since the variable, x, is raised to the first power. Solve for x. Solution x 4 = 7 Add 4 to both sides. x = Simplify. x = 11 There is one solution. Check ( 11) 4 7 Example: Solve the equation. 5y = 40 = Substitute x =11 into the original equation and simplify the results. 7= 7 The statement is true so x = 11 is the solution. This is a linear equation since the variable, y, is raised to the first power. Solve for y Solution 5y = 40 5y 5 = 40 5 y = 8 Check 58 ( ) 40 Divide by 5. Note: Dividing by 5 is equivalent to multiplying by 1/5. There is one solution. x =11 results. The statement is true so x = 8 is the solution = Substitute into the original equation and simplify the

8 Page 8 of 13 Example: Solve the equation. 3x + 2 = 17 This is a linear equation since the variable, x, is raised to the first power. Solve for x. Solution 3x + 2 = 17 Subtract 2 from both sides. 3x = x = 15 3x 3 = 15 Divide both sides by 3. 3 x = 5 There is one solution. Check ( ) 35+ 2= 17 Substitute x = 5 into the original equation and simplify the = 17 results. 17 = 17 The statement is true so x = 5 is the solution. The previous examples have all been in the most basic form. As the problems become more involved you may want to follow the procedure outlined below. Steps used to Solve a Linear Equation 1. Simplify the algebraic expressions on each side of the equation (remove grouping symbols and combine like terms). 2. Collect all the terms with the variable on one side of the equation and all the constants (numerical terms) on the other side. This isolates the variable. 3. Divide both sides by the coefficient of the variable. (Multiplying by the reciprocal of the coefficient of the variable gives the same result.) 4. Check or verify the solution by substituting the value into the original equation.

9 Page 9 of 13 Example: Solve the equation. 2( x 4) 5x + 2 = 3 This is a linear equation since the variable, x, is raised to the first power. Solve for x Solution 2( x 4) 5x + 2 = 3 Use the distributive property. 2x 8 5x + 2 = 3 3x 6 = 3 Collect the constants on one side, so add 6 to both sides. 3x = 3+ 6 Simplify. 3x = 3 3x 3 = 3 Divide both sides by 3, the coefficient of x. 3 x = 1 There is one possible solution. Verification 2( x 4) 5x + 2 = 3 Original Equation 2( 1 4) 5( 1)+ 2 = 3 Substitute 1 for x in the original equation. 2( 5) 5( 1)+ 2 = 3 Simplify inside the parentheses = 3 Perform multiplication operations. 3 = 3 The statement is true so x = 1 is a solution. Example: Solve the equation. 5( 2z 8) 2 = 5( z 3)+ 3 (Verification will be omitted for this problem.) This is a linear equation since the variable, z, is raised to the first power. Solve for z. Solution 52z ( 8) 2 = 5( z 3)+ 3 10z 40 2 = 5x x 42 = 5x 12 10x 42 5x = 5x 12 5x 5x 42 = 12 5x = Use the distributive property. Subtract 5x from both sides to collect the variables on one side. Add 42 to both sides to collect the constants on the opposite side of the variables. 5x = 30 Divide both sides by 5. x = 6 Check the solution on your own.

10 Page 10 of 13 Example: Solve the equation. 3x 4 3 = x (Verification will be omitted for this problem.) Solution This is a linear equation involving fractions. Begin by multiplying both sides of the equation by the least common denominator. 4 3x 4 3 = x x 4 3 = x Multiply both sides by 4 4 3x = 4 x Use the distributive property. 3x 12 = 2x + 8 Simplify the fractions. 3x 12 2x = 2x + 8 2x Subtract 2x from both sides to collect the variables on one side. x 12 = 8 x = Add 12 to both sides to collect the constants on the opposite side of the variables. x = 20 Linear Equations can have one, none, or many solutions. Thus far all of our examples have had a single solution. It is also possible that a linear equation will have no solution or all real numbers as the solution. When solving an equation with no solution, the variable will be eliminated as you attempt to collect the variables on one side and the constants on the other side. This will result in a false statement, such as 5 = 8 or 3= 1. Thus, when you obtain a false statement there are no solutions to the linear equation. When solving an equation with all real numbers as the solution, you will eliminate the variable as you collect the variables on one side and the constants on the other side. Instead of a false statement you will obtain a true statement, such as 7 = 7. Thus, when you obtain a true statement, an identity, any real number will satisfy the linear equation.

11 Page 11 of 13 Example: Solve the equation. 2( 2x 1) 3 = 4( x + 2) This is a linear equation in the variable x. Solve for x. Solution 22x ( 1) 3 = 4( x + 2) 4x 2 3 = 4x + 8 4x 5 = 4x + 8 4x 5 4x = 4x + 8 4x 5 = 8 No Solution Use the distributive property. Subtract 4x from both sides. The variable is eliminated. The statement is false. To verify there is no solution pick any real number for x. Substitute the value into the original equation. Verification 2( 2( 5) 1) 3= 4( 5+ 2) Substitute 5 for x in the original equation and simplify the results. Choose a second x value like x = 0 and check again. 210 ( 1) 3= 47 ( ) 29 ( ) 28 The statement is false which verifies x = 5 is not a solution. Any value you choose would result in a false statement so there are no solutions. Example: Solve the equation. 5y 3( y 1)= 2( y + 3) 3 This is a linear equation in the variable y. Solve for y. Solution 5y 3( y 1)= 2( y + 3) 3 5y 3y + 3 = 2y y + 3 = 2y + 3 2y + 3 2y = 2y + 3 2y Use the distributive property. Combine like terms Subtract 2y from each side. 3 = 3 The variable y is eliminated. The statement is true. y y Any real number will satisfy the equation. The { is a real number} solution is the set of real numbers. ( y ) To verify any real number is a solution pick any real number for x. Substitute the value into the original equation. This is shown for y = 4 on the next page.

12 Page 12 of 13 Verification ( ) ( ) ( ) 20 9 = 2( 7) = Substitute 4 for y in the original equation and simplify the results. Choose a second y like y = 0 or y = 1 and check again. 11 = 11 The statement is true which verifies y = 4 is a solution. Any value you choose would result in a true statement so y is any real number. Graphs of Linear Functions The graph of a linear function f( x)= mx + b is a line with slope m and y-intercept b. This format is called slope-intercept form. It is also written y = mx +b. The slope of a line through the points ( x 1, y 1 ) and ( x 2, y 2 ) is given by the formula m = y 2 y 1. x 2 x 1 Example: Determine the slope of the line through the points ( 5, 8) and ( 13, 4) When substituting into the slope formula it does not matter which point is chosen to be ( x 1, y 1 ). In this example we will let ( x 1, y 1 ) be ( 13, 4) and we will let ( x 2, y 2 ) be ( 5, 8). Solution m = 8 ( 4) 5 13 m = 12 8 m = 3 2 Substitute into the slope formula. Perform the subtraction operation in the numerator and denominator. Reduce the fraction.

13 Page 13 of 13 Example: Write the linear equation 3x + 4y = 6 in slope-intercept form. y = mx+ b or f ( x) = mx+ b To write the equation in slope-intercept form solve the equation for y. Solution 3x + 4y = 6 4y = 3x + 6 y = 3x y = 3 4 x f ( x) = x+ 4 2 or Subtract 3x from both sides. Divide both sides by 4. To obtain slope intercept form, separate the right side into two terms and simplify fractions if possible. Slope intercept form using y = mx+ b Slope intercept form using function notation Example: Determine the slope intercept form for the equation of the line through the points ( 12, 6) and ( 9, 8). To find the equation of the line determine the slope, m, and the y-intercept, b. Solution = = 2 3 ( ) be ( 12, 6) and Determine the slope. Let x 2, y 2 ( x 1, y 1 ) be ( 9, 8). Remember to reduce the fraction. 8 = 2 3 ()+ 9 b Determine the y-intercept. Substitute the slope of 2 3 and the 8 = 6 + b 2 = b ( ) into the linear function formula. Solve for b. point 9, 8 Note: Either point can be used to find b. f()= x 2 3 x + 2 State the equation using function notation.

### MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

### 2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

### Algebra Cheat Sheets

Sheets Algebra Cheat Sheets provide you with a tool for teaching your students note-taking, problem-solving, and organizational skills in the context of algebra lessons. These sheets teach the concepts

### 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

### 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

### 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

### 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

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### 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

### 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

### 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,

### 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

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Mathematics Placement

Mathematics Placement The ACT COMPASS math test is a self-adaptive test, which potentially tests students within four different levels of math including pre-algebra, algebra, college algebra, and trigonometry.

### 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

### 2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

### What are the place values to the left of the decimal point and their associated powers of ten?

The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything

### 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

### 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

### 1.3 Polynomials and Factoring

1.3 Polynomials and Factoring Polynomials Constant: a number, such as 5 or 27 Variable: a letter or symbol that represents a value. Term: a constant, variable, or the product or a constant and variable.

### IV. ALGEBRAIC CONCEPTS

IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### 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

### Algebra 2 PreAP. Name Period

Algebra 2 PreAP Name Period IMPORTANT INSTRUCTIONS FOR STUDENTS!!! We understand that students come to Algebra II with different strengths and needs. For this reason, students have options for completing

### Fractions and Linear Equations

Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

### 3. Solve the equation containing only one variable for that variable.

Question : How do you solve a system of linear equations? There are two basic strategies for solving a system of two linear equations and two variables. In each strategy, one of the variables is eliminated

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Review of Intermediate Algebra Content

Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6

Factoring the trinomial ax 2 + bx + c when a = 1 A trinomial in the form x 2 + bx + c can be factored to equal (x + m)(x + n) when the product of m x n equals c and the sum of m + n equals b. (Note: the

### Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

### COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1

10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions

### Florida Algebra 1 End-of-Course Assessment Item Bank, Polk County School District

Benchmark: MA.912.A.2.3; Describe the concept of a function, use function notation, determine whether a given relation is a function, and link equations to functions. Also assesses MA.912.A.2.13; Solve

### The Properties of Signed Numbers Section 1.2 The Commutative Properties If a and b are any numbers,

1 Summary DEFINITION/PROCEDURE EXAMPLE REFERENCE From Arithmetic to Algebra Section 1.1 Addition x y means the sum of x and y or x plus y. Some other words The sum of x and 5 is x 5. indicating addition

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level 2

COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what

8. Radicals - Rational Exponents Objective: Convert between radical notation and exponential notation and simplify expressions with rational exponents using the properties of exponents. When we simplify

### 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

### MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)

MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,

### SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS

(Section 0.6: Polynomial, Rational, and Algebraic Expressions) 0.6.1 SECTION 0.6: POLYNOMIAL, RATIONAL, AND ALGEBRAIC EXPRESSIONS LEARNING OBJECTIVES Be able to identify polynomial, rational, and algebraic

### 1.3 Algebraic Expressions

1.3 Algebraic Expressions A polynomial is an expression of the form: a n x n + a n 1 x n 1 +... + a 2 x 2 + a 1 x + a 0 The numbers a 1, a 2,..., a n are called coefficients. Each of the separate parts,

### Graphing - Parallel and Perpendicular Lines

. Graphing - Parallel and Perpendicular Lines Objective: Identify the equation of a line given a parallel or perpendicular line. There is an interesting connection between the slope of lines that are parallel

### Absolute Value Equations and Inequalities

. Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

### 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

### A Quick Algebra Review

1. Simplifying Epressions. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Eponents 9. Quadratics 10. Rationals 11. Radicals

### Linear Equations in One Variable

Linear Equations in One Variable MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this section we will learn how to: Recognize and combine like terms. Solve

### Make sure you look at the reminders or examples before each set of problems to jog your memory! Solve

Name Date Make sure you look at the reminders or examples before each set of problems to jog your memory! I. Solving Linear Equations 1. Eliminate parentheses. Combine like terms 3. Eliminate terms by

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

### 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

### 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

### SUNY ECC. ACCUPLACER Preparation Workshop. Algebra Skills

SUNY ECC ACCUPLACER Preparation Workshop Algebra Skills Gail A. Butler Ph.D. Evaluating Algebraic Epressions Substitute the value (#) in place of the letter (variable). Follow order of operations!!! E)

### 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

### 1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

### MATH 90 CHAPTER 1 Name:.

MATH 90 CHAPTER 1 Name:. 1.1 Introduction to Algebra Need To Know What are Algebraic Expressions? Translating Expressions Equations What is Algebra? They say the only thing that stays the same is change.

### 6.1 The Greatest Common Factor; Factoring by Grouping

386 CHAPTER 6 Factoring and Applications 6.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### 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,

### PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71. Applications. F = mc + b.

PRIMARY CONTENT MODULE Algebra I -Linear Equations & Inequalities T-71 Applications The formula y = mx + b sometimes appears with different symbols. For example, instead of x, we could use the letter C.

### 7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

### 2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

1.6 A LIBRARY OF PARENT FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Identify and graph linear and squaring functions. Identify and graph cubic, square root, and reciprocal

### Algebra 1 If you are okay with that placement then you have no further action to take Algebra 1 Portion of the Math Placement Test

Dear Parents, Based on the results of the High School Placement Test (HSPT), your child should forecast to take Algebra 1 this fall. If you are okay with that placement then you have no further action

### IOWA End-of-Course Assessment Programs. Released Items ALGEBRA I. Copyright 2010 by The University of Iowa.

IOWA End-of-Course Assessment Programs Released Items Copyright 2010 by The University of Iowa. ALGEBRA I 1 Sally works as a car salesperson and earns a monthly salary of \$2,000. She also earns \$500 for

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

### 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.

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III

Alum Rock Elementary Union School District Algebra I Study Guide for Benchmark III Name Date Adding and Subtracting Polynomials Algebra Standard 10.0 A polynomial is a sum of one ore more monomials. Polynomial

### Graphing - Slope-Intercept Form

2.3 Graphing - Slope-Intercept Form Objective: Give the equation of a line with a known slope and y-intercept. When graphing a line we found one method we could use is to make a table of values. However,

### Mathematics Common Core Sample Questions

New York State Testing Program Mathematics Common Core Sample Questions Grade The materials contained herein are intended for use by New York State teachers. Permission is hereby granted to teachers and

### Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions 1 Created January 17, 2006

MTH 092 College Algebra Essex County College Division of Mathematics Sample Review Questions Created January 7, 2006 Math 092, Elementary Algebra, covers the mathematical content listed below. In order

### 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

### 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

### Algebra I Teacher Notes Expressions, Equations, and Formulas Review

Big Ideas Write and evaluate algebraic expressions Use expressions to write equations and inequalities Solve equations Represent functions as verbal rules, equations, tables and graphs Review these concepts

### Answers to Basic Algebra Review

Answers to Basic Algebra Review 1. -1.1 Follow the sign rules when adding and subtracting: If the numbers have the same sign, add them together and keep the sign. If the numbers have different signs, subtract

### Five 5. Rational Expressions and Equations C H A P T E R

Five C H A P T E R Rational Epressions and Equations. Rational Epressions and Functions. Multiplication and Division of Rational Epressions. Addition and Subtraction of Rational Epressions.4 Comple Fractions.

### 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

### MATH 10034 Fundamental Mathematics IV

MATH 0034 Fundamental Mathematics IV http://www.math.kent.edu/ebooks/0034/funmath4.pdf Department of Mathematical Sciences Kent State University January 2, 2009 ii Contents To the Instructor v Polynomials.

### MATD 0390 - Intermediate Algebra Review for Pretest

MATD 090 - Intermediate Algebra Review for Pretest. Evaluate: a) - b) - c) (-) d) 0. Evaluate: [ - ( - )]. Evaluate: - -(-7) + (-8). Evaluate: - - + [6 - ( - 9)]. Simplify: [x - (x - )] 6. Solve: -(x +

### Math 25 Activity 6: Factoring Advanced

Instructor! Math 25 Activity 6: Factoring Advanced Last week we looked at greatest common factors and the basics of factoring out the GCF. In this second activity, we will discuss factoring more difficult

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### The Greatest Common Factor; Factoring by Grouping

296 CHAPTER 5 Factoring and Applications 5.1 The Greatest Common Factor; Factoring by Grouping OBJECTIVES 1 Find the greatest common factor of a list of terms. 2 Factor out the greatest common factor.

### 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

### Veterans Upward Bound Algebra I Concepts - Honors

Veterans Upward Bound Algebra I Concepts - Honors Brenda Meery Kaitlyn Spong Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) www.ck12.org Chapter 6. Factoring CHAPTER

### Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### Factoring (pp. 1 of 4)

Factoring (pp. 1 of 4) Algebra Review Try these items from middle school math. A) What numbers are the factors of 4? B) Write down the prime factorization of 7. C) 6 Simplify 48 using the greatest common

### EAP/GWL Rev. 1/2011 Page 1 of 5. Factoring a polynomial is the process of writing it as the product of two or more polynomial factors.

EAP/GWL Rev. 1/2011 Page 1 of 5 Factoring a polynomial is the process of writing it as the product of two or more polynomial factors. Example: Set the factors of a polynomial equation (as opposed to an

### 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

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### POLYNOMIALS and FACTORING

POLYNOMIALS and FACTORING Exponents ( days); 1. Evaluate exponential expressions. Use the product rule for exponents, 1. How do you remember the rules for exponents?. How do you decide which rule to use

### How To Understand And Solve Algebraic Equations

College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGraw-Hill, 2008, ISBN: 978-0-07-286738-1 Course Description This course provides

### SIMPLIFYING ALGEBRAIC FRACTIONS

Tallahassee Community College 5 SIMPLIFYING ALGEBRAIC FRACTIONS In arithmetic, you learned that a fraction is in simplest form if the Greatest Common Factor (GCF) of the numerator and the denominator is

### SAT Math Facts & Formulas Review Quiz

Test your knowledge of SAT math facts, formulas, and vocabulary with the following quiz. Some questions are more challenging, just like a few of the questions that you ll encounter on the SAT; these questions