Precalculus: Linear Equations Practice Problems. Questions. 1. Solve for x when 2 3 x = 1 15 x Solve for x when x 2 + x 5 = 7 10.

Size: px
Start display at page:

Download "Precalculus: Linear Equations Practice Problems. Questions. 1. Solve for x when 2 3 x = 1 15 x Solve for x when x 2 + x 5 = 7 10."

Transcription

1 Questions. Solve for x when 3 x = 5 x Solve for x when x + x 5 = Solve for x when 0 3 x = x. 4. Is 4 a solution to (y ) + = 3 (3y 4)? 8 5. Solve for x when 4 5 x 3 = 3x Solve for x when + 5(x ) = x + 3 7x. 7. Solve for x when 9(x + 3) 6 = 4 x 3 + x. 8. Sketch y = x +. Find the value of y when x = 0, x =, and x =. 9. Sketch y = x 5. Find the value of y when x = 0, x =, and x = Sketch y = 3x +. Find the value of y when x =, x = 0, and x =.. Sketch 4x + 3y =.. Sketch 3x + y = Sketch y = 6 x. 4. Sketch x 6 = y. 5. Sketch y = 3y. 6. Sketch x + 9 = 5x. 7. Sketch x + 5y =. 8. Find the slope of the straight line that passes through the points (4, ) and (6, 7). 9. Find the slope of the straight line that passes through the points (, ) and (5, 4). 0. Find the slope of the straight line that passes through the points ( 6, 5) and (, 7).. Write the equation for a straight line in slope-intercept form with slope m = 3 and y-intercept (0, 5).. Write the equation for a straight line in slope-intercept form with slope m = 5 and y-intercept (0, 6). 3. Write the equation for a straight line in slope-intercept form with slope m = 3 and y-intercept (0, /). 4. Sketch the straight line y = mx + b where m = 3 and b =. 5. Sketch the straight line y = mx + b where m = 3 and b = Sketch the straight line y = 3x. 7. A line has a slope of. What is the slope of a line parallel to it? What is the slope of a line perpendicular to it? 4 8. A line has equation y = 3 x 5. What is the slope of a line parallel to it? What is the slope of a line perpendicular 5 to it? Page of 0

2 9. During the years from 980 to 005 the total income for the U.S. federal budget can be approximated by the equation y = 4(4x + 35), where x is the number of years since 980 and y is the amount of money in billions of dollars (source: U.S. Office of Management and Budget). Write the equation in slope-intercept form. Find the slope and y-intercept. What is the meaning of the slope in this situation? 30. Find the equation of the line that passes through the point (5, 3) and has slope m = 5. ( 3. Find the equation of the line that passes through the points, 5 ) ( and 3, 3 ). 6 ( 3 3. Find the equation of the line that passes through the points (, 0) and, ). 33. Find the equation of the line that passes through the point (4, 3) and has slope m =. 34. Find the equation of the line that passes through the points (, 8) and (, 4). 35. Write the equation of the line given below. 37. Write the equation of the line given below. 36. Write the equation of the line given below. 38. Write the equation of the line given below. Page of 0

3 Solutions. The LCD (lowest common denominator) is 5, so multiply the equation by 5 to remove the fractions. 3 x = 5 x + 3 ( ) ( x = 5 5 x + 3 ) 5 0x = 5 5 x distribute! 0x = x + 9 simplify. LCD is 0. 0x x = x + 9 x addition principle 9x = 9 simplify 9 9x = 9 multiplication principle 9 x = simplify x + x 5 = 7 0 ( x 0 + x ) = x + 0 x 5 = 7 3. LCD is 6. 5x + x = x = 7 7 7x = 7 7 x = 0 3 x = x 6 (0 3 ) x = 6 x x = 3x 0 x = 3x 0 x + x = 3x + x 0 = 5x 5 0 = 5 5x 4 = x Page 3 of 0

4 4. You could substitute y = 4 to check, but I am going to solve it instead. LCD is 8. (y ) + = 3 (3y 4) ( ) 8 8 (y ) + = 8 3 (3y 4) 8 8 (y ) + 8 = 3(3y 4) 4(y ) + 6 = 9y 4y = 9y 4y + 8 = 9y 4y + 8 9y 8 = 9y 9y 8 5y = 0 5 ( 5y) = 5 ( 0) y = 4 5. LCD is x 3 = 3x + ) ( 4 5 x 3 = 30 3x x 30 3 = 30 (3x + ) Note in above I wrote 3x + 6. as 4x 0 = 5 (3x + ) 4x 0 = 45x + 5 4x 0 45x + 0 = 45x x + 0 x = 35 ( x) = 35 x = 35 = (x ) = x + 3 7x + 5x 0 = 5x + 3 5x 9 5x = 5x + 3 5x 9 = 3 (3x + ). Doing this helps reduce errors! We have to interpret what we have found. Since 9 never equals 3, the equation is never true no matter what value of x we put in. This means the equation has no solution. Page 4 of 0

5 7. 9(x + 3) 6 = 4 x 3 + x 9x = + 9x 9x + = + 9x 9x + 9x = + 9x 9x = We have to interpret what we have found. Since is always equal to, the equation is true for any value of x that we try. Therefore, there are an infinite number of solutions. 8. y = x + When x = 0 y = (0) + =, so the ordered pair is (0, ). When x = y = ( ) + = 5, so the ordered pair is (, 5). When x = y = () + =, so the ordered pair is (, ). 9. y = x 5 When x = 0 y = (0) 5 = 5, so the ordered pair is (0, 5). When x = y = () 5 =, so the ordered pair is (, ). When x = 4 y = (4) 5 = 3, so the ordered pair is (4, 3). 0. y = 3x + When x = y = 3( ) + =, so the ordered pair is (, ). When x = 0 y = 3(0) + =, so the ordered pair is (0, ). When x = y = 3() + = 5, so the ordered pair is (, 5). Page 5 of 0

6 . 4x + 3y = When x = 0 4(0) + 3y = y = 4 so the ordered pair is (0, 4). When y = 0 4x + 3(0) = x = 3 so the ordered pair is (3, 0). When x = 4() + 3y = y = 8/3 so the ordered pair is (, 8/3).. 3x + y = 6 When x = 0 3(0) + y = 6 y = 3 so the ordered pair is (0, 3). When y = 0 3x + (0) = 6 x = so the ordered pair is (, 0). When x = 3() + y = 6 y = 3/ so the ordered pair is (, 3/). 3. y = 6 x When x = 0 y = 6 (0) y = 6 so the ordered pair is (0, 6). When y = 0 (0) = 6 x x = 3 so the ordered pair is (3, 0). When x = y = 6 () y = 4 so the ordered pair is (, 4). 4. x 6 = y When x = 0 (0) 6 = y y = 3 so the ordered pair is (0, 3). When y = 0 x 6 = (0) x = 6 so the ordered pair is (6, 0). When x = 8 (8) 6 = y y = so the ordered pair is (8, ). Page 6 of 0

7 5. y = 3y. There is no x in the equation. Simplification shows this is a horizontal line, y =. 6. x + 9 = 5x. There is no y in the equation. Simplification shows this is a vertical line, x = x + 5y = x + 5y = 0 When x = 0 (0) + 5y = 0 y = so the ordered pair is (0, ). When y = 0 x + 5(0) = 0 x = 5 so the ordered pair is ( 5, 0). When x = 5 (5) + 5y = 0 y = 4 so the ordered pair is (5, 4). 8. slope = y x = = 6 = slope = y x = 4 5 = = slope = y 5 ( 7) = = x 6 8 = 4.. y = 3 x y = 5x y = 3 x y = 3 x. Page 7 of 0

8 5. y = 3 x y = 3x. Intercept is b = 0. Slope is m = rise run = Parallel: 4. Perpendicular: Parallel: 3 5. Perpendicular: y = 4(4x + 35) = 56x slope = 56 and y-intercept is (0, 490). The slope is the amount of increase in income of the federal budget in billions of dollars per year. Aside: This equation is not as good as it could be, since x represents the number of years since 980. The equation would be improved if the independent variable represented the year. We can make this change by introducing a change in variables. Let z be the year. Then z = x. Therefore, x = z 980. The equation becomes y = 56x y = 56(z 980) = 56z 0, 390 What was the federal budget in 987? Answer: y = 56z 0, 390 = 56(987) 0, 390 = 88 billion dollars. This is the same answer you get if you use y = 56x with x = Use the slope-point equation of a line. y y = m(x x ) y ( 3) = (x 5) 5 y + 3 = 5 x + y = 5 x 3. slope = y 5 x = = ( ) 4 6 = ( 4 ) = 6 3. Page 8 of 0

9 Now use the slope-point equation of a line. y y = m(x x ) y 5 6 = (x ) 3 y 5 6 = 3 x 3 y = 3 x y = 3 x y = 3 x y = 3 x + 3. slope = y x = 0 3 ( ) = ( = ) Now use the slope-point equation of a line. y y = m(x x ) y 0 = (x ) y = x Use the slope-point equation of a line. y y = m(x x ) y (3) = (x 4) y 3 = x + 8 y = x y = x + ( ) =. 34. slope = y 8 ( 4) = = = 6 x = 6. Now use the slope-point equation of a line. y y = m(x x ) y ( 8) = 6(x ) y + 8 = 6x + 6 y = 6x y = 6x Page 9 of 0

10 35. You need to be able to read these off the sketch. Look for two points that the line crosses a grid line intersection. Two points: (0, ) and (3, ). Rise =, Run = 3. slope = rise run = 3 = 3. y-intercept b =. 37. This is a horizontal line, so it s equation is just y =. y = mx + b y = 3 x Two points: (0, 4) and (3, ). Rise =, Run = 3. slope = rise run = 3. y-intercept b = 4. y = mx + b y = 3 x Two points: (0, 0) and (, 3). Rise = 3, Run =. slope = rise run = 3. y-intercept b = 0. y = mx + b y = 3 x. Page 0 of 0

GUIDED NOTES 4.1 LINEAR FUNCTIONS

GUIDED NOTES 4.1 LINEAR FUNCTIONS GUIDED NOTES 4.1 LINEAR FUNCTIONS LEARNING OBJECTIVES In this section, you will: Represent a linear function. Determine whether a linear function is increasing, decreasing, or constant. Interpret slope

More information

1.1 : (The Slope of a straight Line)

1.1 : (The Slope of a straight Line) 1.1 : (The Slope of a straight Line) Equations of Nonvertical Lines: A nonvertical line L has an equation of the form y mx b. The number m is called the slope of L and the point (0, b) is called the y-intercept.

More information

Twitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Find the point of intersection for each pair of lines: a) y = 4x + 7 and 5y = 2x 1 b) y = 5x 1 and 3x + 7y = 11 c) 2x 5y =

More information

Section 3.4 Writing the Equation of a Line

Section 3.4 Writing the Equation of a Line Chapter Linear Equations and Functions Section.4 Writing the Equation of a Line Writing Equations of Lines Critical to a thorough understanding of linear equations and functions is the ability to write

More information

GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE

GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE GUIDED NOTES 2.2 LINEAR EQUATIONS IN ONE VARIABLE LEARNING OBJECTIVES In this section, you will: Solve equations in one variable algebraically. Solve a rational equation. Find a linear equation. Given

More information

Characteristics of Linear Functions (pp. 1 of 8)

Characteristics of Linear Functions (pp. 1 of 8) Characteristics of Linear Functions (pp. 1 of 8) Algebra 2 Parent Function Table Linear Parent Function: x y y = Domain: Range: What patterns do you observe in the table and graph of the linear parent

More information

Section 1.6. Functions

Section 1.6. Functions Section 1.6 Functions Definitions Relation, Domain, Range, and Function The table describes a relationship between the variables x and y. This relationship is also described graphically. x y 3 2 4 1 5

More information

Math M111: Lecture Notes For Chapter 3

Math M111: Lecture Notes For Chapter 3 Section 3.1: Math M111: Lecture Notes For Chapter 3 Note: Make sure you already printed the graphing papers Plotting Points, Quadrant s signs, x-intercepts and y-intercepts Example 1: Plot the following

More information

Sect The Slope-Intercept Form

Sect The Slope-Intercept Form 0 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

Unit 7: It s in the System

Unit 7: It s in the System Unit 7: It s in the System Investigation 1: Linear Equations with Two Variables I can convert between standard and slope intercept forms, and graph systems of equations. Solving equations is one of the

More information

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables

Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Sections 8.1 & 8.2 Systems of Linear Equations in Two Variables Department of Mathematics Porterville College September 7, 2014 Systems of Linear Equations in Two Variables Learning Objectives: Solve Systems

More information

Lesson 3-1: Solving Linear Systems by Graphing

Lesson 3-1: Solving Linear Systems by Graphing For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we

More information

Matrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form.

Matrices. A matrix is a method of writing a set of numbers using rows and columns. Cells in a matrix can be referenced in the form. Matrices A matrix is a method of writing a set of numbers using rows and columns. 1 2 3 4 3 2 1 5 7 2 5 4 2 0 5 10 12 8 4 9 25 30 1 1 Reading Information from a Matrix Cells in a matrix can be referenced

More information

1 Numbers, Sets, Algebraic Expressions

1 Numbers, Sets, Algebraic Expressions AAU - Business Mathematics I Lecture #1, February 27, 2010 1 Numbers, Sets, Algebraic Expressions 1.1 Constants, Variables, and Sets A constant is something that does not change, over time or otherwise:

More information

Edexcel New GCE A Level Maths workbook

Edexcel New GCE A Level Maths workbook Edexcel New GCE A Level Maths workbook Straight line graphs Parallel and Perpendicular lines. Edited by: K V Kumaran kumarmaths.weebly.com Straight line graphs A LEVEL LINKS Scheme of work: a. Straight-line

More information

( ) c. m = 0, 1 2, 3 4

( ) c. m = 0, 1 2, 3 4 G Linear Functions Probably the most important concept from precalculus that is required for differential calculus is that of linear functions The formulas you need to know backwards and forwards are:

More information

Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review

Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review Math 0312 Intermediate Algebra Chapter 1 and 2 Test Review Name Date MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation for y. 1)

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

SKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution.

SKILL BUILDER TEN. Graphs of Linear Equations with Two Variables. If x = 2 then y = = = 7 and (2, 7) is a solution. SKILL BUILDER TEN Graphs of Linear Equations with Two Variables A first degree equation is called a linear equation, since its graph is a straight line. In a linear equation, each term is a constant or

More information

Graphical Solutions of Linear Systems

Graphical Solutions of Linear Systems Graphical Solutions of Linear Systems Consistent System (At least one solution) Inconsistent System (No Solution) Independent (One solution) Dependent (Infinite many solutions) Parallel Lines Equations

More information

The Circle. 1 The equation of a circle. Nikos Apostolakis. Spring 2017

The Circle. 1 The equation of a circle. Nikos Apostolakis. Spring 2017 The Circle Nikos Apostolakis Spring 017 1 The equation of a circle Definition 1. Given a point C in the plane and a positive number r, the circle with center C and radius r, is the locus of points that

More information

Study Guide for Math 095

Study Guide for Math 095 Study Guide for Math 095 David G. Radcliffe November 7, 1994 1 The Real Number System Writing a fraction in lowest terms. 1. Find the largest number that will divide into both the numerator and the denominator.

More information

Consistent and Dependent

Consistent and Dependent Graphing a System of Equations System of Equations: Consists of two equations. The solution to the system is an ordered pair that satisfies both equations. There are three methods to solving a system;

More information

SOLUTIONS FOR PROBLEMS 1-30

SOLUTIONS FOR PROBLEMS 1-30 . Answer: 5 Evaluate x x + 9 for x SOLUTIONS FOR PROBLEMS - 0 When substituting x in x be sure to do the exponent before the multiplication by to get (). + 9 5 + When multiplying ( ) so that ( 7) ( ).

More information

3.3 Linear Equations in Standard Form

3.3 Linear Equations in Standard Form 3.3 Linear Equations in Standard Form Learning Objectives Write equivalent equations in standard form. Find the slope and y intercept from an equation in standard form. Write equations in standard form

More information

ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET

ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET NAME ADVANCED/HONORS ALGEBRA 2 - SUMMER PACKET Part I. Order of Operations (PEMDAS) Parenthesis and other grouping symbols. Exponential expressions. Multiplication & Division. Addition & Subtraction. Tutorial:

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

Chapter 1-2 Add and Subtract Integers

Chapter 1-2 Add and Subtract Integers Chapter 1-2 Add and Subtract Integers Absolute Value of a number is its distance from zero on the number line. 5 = 5 and 5 = 5 Adding Numbers with the Same Sign: Add the absolute values and use the sign

More information

Brooklyn College Department of Mathematics. Precalculus. Preparatory Workbook. Spring Sandra Kingan

Brooklyn College Department of Mathematics. Precalculus. Preparatory Workbook. Spring Sandra Kingan Brooklyn College Department of Mathematics Precalculus Preparatory Workbook Spring 0 Sandra Kingan Supported by the CUNY Office of Academic Affairs through funding for the Gap Project CONTENTS. Review

More information

Algebra 1 (cp) Midterm Review Name: Date: Period:

Algebra 1 (cp) Midterm Review Name: Date: Period: Algebra 1 (cp) Midterm Review Name: Date: Period: Chapter 1 1. Evaluate the variable expression when j 4. j 44 [1] 2. Evaluate the variable expression when j 4. 24 j [2] 3. Find the perimeter of the rectangle.

More information

ACCUPLACER MATH 0311 OR MATH 0120

ACCUPLACER MATH 0311 OR MATH 0120 The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 0 OR MATH 00 http://www.academics.utep.edu/tlc MATH 0 OR MATH 00 Page Factoring Factoring Eercises 8 Factoring Answer to Eercises

More information

Systems of Equations and Inequalities. College Algebra

Systems of Equations and Inequalities. College Algebra Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system

More information

Average Rate of Change & Slope of a Line MATH 092

Average Rate of Change & Slope of a Line MATH 092 Average Rate of Change Average Rate of Change & Slope of a Line MATH 092 Functions are used to model the way one quantity changes with respect to another quantity. For instance, how does the distance traveled

More information

a) Graph the equation by the intercepts method. Clearly label the axes and the intercepts. b) Find the slope of the line.

a) Graph the equation by the intercepts method. Clearly label the axes and the intercepts. b) Find the slope of the line. Math 71 Spring 2009 TEST 1 @ 120 points Name: Write in a neat and organized fashion. Write your complete solutions on SEPARATE PAPER. You should use a pencil. For an exercise to be complete there needs

More information

Unit 1 PreCalculus Review & Limits

Unit 1 PreCalculus Review & Limits 1 Unit 1 PreCalculus Review & Limits Factoring: Remove common factors first Terms - Difference of Squares a b a b a b - Sum of Cubes ( )( ) a b a b a ab b 3 3 - Difference of Cubes a b a b a ab b 3 3 3

More information

Equations With Two or More Variables

Equations With Two or More Variables ! Equations With Two or More Variables You have done a lot of work with relationships involving two related variables. However, many real-world relationships involve three or more variables. For example,

More information

1.2 Graphs and Lines. Cartesian Coordinate System

1.2 Graphs and Lines. Cartesian Coordinate System 1.2 Graphs and Lines Cartesian Coordinate System Note that there is a one-to-one correspondence between the points in a plane and the elements in the set of all ordered pairs (a, b) of real numbers. Graphs

More information

Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator.

Adding and Subtracting Rational Expressions. Add and subtract rational expressions with the same denominator. Chapter 7 Section 7. Objectives Adding and Subtracting Rational Expressions 1 3 Add and subtract rational expressions with the same denominator. Find a least common denominator. Add and subtract rational

More information

You try: What is the equation of the line on the graph below? What is the equation of the line on the graph below?

You try: What is the equation of the line on the graph below? What is the equation of the line on the graph below? 1 What is the equation of the line on the graph below? 2 3 1a What is the equation of the line on the graph below? y-intercept Solution: To write an equation in slope-intercept form, identify the slope

More information

Reteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class

Reteach 2-3. Graphing Linear Functions. 22 Holt Algebra 2. Name Date Class -3 Graphing Linear Functions Use intercepts to sketch the graph of the function 3x 6y 1. The x-intercept is where the graph crosses the x-axis. To find the x-intercept, set y 0 and solve for x. 3x 6y 1

More information

TEST 150 points Write neatly. Show all work. Write all responses on separate paper. Clearly label the exercises.

TEST 150 points Write neatly. Show all work. Write all responses on separate paper. Clearly label the exercises. Math 130 Fall 007 Name: TEST # @ 150 points Write neatly. Show all work. Write all responses on separate paper. Clearly label the exercises. 1. Consider the polynomial function 4 3 f ( x) = x 19x + 57x

More information

OBJECTIVES UNIT 1. Lesson 1.0

OBJECTIVES UNIT 1. Lesson 1.0 OBJECTIVES UNIT 1 Lesson 1.0 1. Define "set," "element," "finite set," and "infinite set," "empty set," and "null set" and give two examples of each term. 2. Define "subset," "universal set," and "disjoint

More information

June If you want, you may scan your assignment and convert it to a.pdf file and it to me.

June If you want, you may scan your assignment and convert it to a.pdf file and  it to me. Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular

More information

Algebra II Summer Packet. Summer Name:

Algebra II Summer Packet. Summer Name: Algebra II Summer Packet Summer 2017 Name: NAME ALGEBRA II & TRIGONOMETRY SUMMER REVIEW PACKET To maintain a high quality program, students entering Algebra II are expected to remember the basics of the

More information

Chapter 1: Precalculus Review

Chapter 1: Precalculus Review : Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,

More information

Geometry Summer Assignment 2018

Geometry Summer Assignment 2018 Geometry Summer Assignment 2018 The following packet contains topics and definitions that you will be required to know in order to succeed in Geometry this year. You are advised to be familiar with each

More information

Pre Algebra. Curriculum (634 topics)

Pre Algebra. Curriculum (634 topics) Pre Algebra This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet curricular needs.

More information

Linear Equations in One Variable *

Linear Equations in One Variable * OpenStax-CNX module: m64441 1 Linear Equations in One Variable * Ramon Emilio Fernandez Based on Linear Equations in One Variable by OpenStax This work is produced by OpenStax-CNX and licensed under the

More information

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra

Module 1: Whole Numbers Module 2: Fractions Module 3: Decimals and Percent Module 4: Real Numbers and Introduction to Algebra Course Title: College Preparatory Mathematics I Prerequisite: Placement with a score below 20 on ACT, below 450 on SAT, or assessing into Basic Applied Mathematics or Basic Algebra using Accuplacer, ASSET

More information

Chapter One: Pre-Geometry

Chapter One: Pre-Geometry Chapter One: Pre-Geometry Index: A: Solving Equations B: Factoring (GCF/DOTS) C: Factoring (Case Two leading into Case One) D: Factoring (Case One) E: Solving Quadratics F: Parallel and Perpendicular Lines

More information

VCE. VCE Maths Methods 1 and 2 Pocket Study Guide

VCE. VCE Maths Methods 1 and 2 Pocket Study Guide VCE VCE Maths Methods 1 and 2 Pocket Study Guide Contents Introduction iv 1 Linear functions 1 2 Quadratic functions 10 3 Cubic functions 16 4 Advanced functions and relations 24 5 Probability and simulation

More information

Chapter 4. Systems of Linear Equations; Matrices. Opening Example. Section 1 Review: Systems of Linear Equations in Two Variables

Chapter 4. Systems of Linear Equations; Matrices. Opening Example. Section 1 Review: Systems of Linear Equations in Two Variables Chapter 4 Systems of Linear Equations; Matrices Section 1 Review: Systems of Linear Equations in Two Variables Opening Example A restaurant serves two types of fish dinners- small for $5.99 and large for

More information

Grade 9 Linear Equations in Two Variables

Grade 9 Linear Equations in Two Variables ID : cn-9-linear-equations-in-two-variables [1] Grade 9 Linear Equations in Two Variables For more such worksheets visit www.edugain.com Answer the questions (1) In the graph of the linear equation 4x

More information

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation?

Algebra Concepts Equation Solving Flow Chart Page 1 of 6. How Do I Solve This Equation? Algebra Concepts Equation Solving Flow Chart Page of 6 How Do I Solve This Equation? First, simplify both sides of the equation as much as possible by: combining like terms, removing parentheses using

More information

Mini Lecture 2.1 Introduction to Functions

Mini Lecture 2.1 Introduction to Functions Mini Lecture.1 Introduction to Functions 1. Find the domain and range of a relation.. Determine whether a relation is a function. 3. Evaluate a function. 1. Find the domain and range of the relation. a.

More information

MATH Spring 2010 Topics per Section

MATH Spring 2010 Topics per Section MATH 101 - Spring 2010 Topics per Section Chapter 1 : These are the topics in ALEKS covered by each Section of the book. Section 1.1 : Section 1.2 : Ordering integers Plotting integers on a number line

More information

B. Linear equations can be written in the form y = mx+b or f(x) = mx+b. which is called: Or Ax + By = C which is called:

B. Linear equations can be written in the form y = mx+b or f(x) = mx+b. which is called: Or Ax + By = C which is called: Math 95, Mod 1, Sec. 3.3 Graphing Linear functions A. Graphing Linear functions E. 1: Graph: f() = 3, g() = 3 + 2 and h() = 3-3 f() g() h() 0-2 2 What happens? B. Linear equations can be written in the

More information

Graphing Review Part 1: Circles, Ellipses and Lines

Graphing Review Part 1: Circles, Ellipses and Lines Graphing Review Part : Circles, Ellipses and Lines Definition The graph of an equation is the set of ordered pairs, (, y), that satisfy the equation We can represent the graph of a function by sketching

More information

NOTES. [Type the document subtitle] Math 0310

NOTES. [Type the document subtitle] Math 0310 NOTES [Type the document subtitle] Math 010 Cartesian Coordinate System We use a rectangular coordinate system to help us map out relations. The coordinate grid has a horizontal axis and a vertical axis.

More information

Systems of Linear Equations: Solving by Graphing

Systems of Linear Equations: Solving by Graphing 8.1 Sstems of Linear Equations: Solving b Graphing 8.1 OBJECTIVE 1. Find the solution(s) for a set of linear equations b graphing NOTE There is no other ordered pair that satisfies both equations. From

More information

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

More information

Geometry/Trig Name: Date: Lesson 1-11 Writing the Equation of a Perpendicular Bisector

Geometry/Trig Name: Date: Lesson 1-11 Writing the Equation of a Perpendicular Bisector Name: Date: Lesson 1-11 Writing the Equation of a Perpendicular Bisector Learning Goals: #14: How do I write the equation of a perpendicular bisector? Warm-up What is the equation of a line that passes

More information

G.3 Forms of Linear Equations in Two Variables

G.3 Forms of Linear Equations in Two Variables section G 2 G. Forms of Linear Equations in Two Variables Forms of Linear Equations Linear equations in two variables can take different forms. Some forms are easier to use for graphing, while others are

More information

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14

Final Exam A Name. 20 i C) Solve the equation by factoring. 4) x2 = x + 30 A) {-5, 6} B) {5, 6} C) {1, 30} D) {-5, -6} -9 ± i 3 14 Final Exam A Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 1 1) x + 3 + 5 x - 3 = 30 (x + 3)(x - 3) 1) A) x -3, 3; B) x -3, 3; {4} C) No restrictions; {3} D)

More information

Homework 4 Solutions, 2/2/7

Homework 4 Solutions, 2/2/7 Homework 4 Solutions, 2/2/7 Question Given that the number a is such that the following limit L exists, determine a and L: x 3 a L x 3 x 2 7x + 2. We notice that the denominator x 2 7x + 2 factorizes as

More information

INTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2

INTERNET MAT 117. Solution for the Review Problems. (1) Let us consider the circle with equation. x 2 + 2x + y 2 + 3y = 3 4. (x + 1) 2 + (y + 3 2 INTERNET MAT 117 Solution for the Review Problems (1) Let us consider the circle with equation x 2 + y 2 + 2x + 3y + 3 4 = 0. (a) Find the standard form of the equation of the circle given above. (i) Group

More information

Lesson 12: Systems of Linear Equations

Lesson 12: Systems of Linear Equations Our final lesson involves the study of systems of linear equations. In this lesson, we examine the relationship between two distinct linear equations. Specifically, we are looking for the point where the

More information

How can you use linear functions of two independent variables to represent problem situations?

How can you use linear functions of two independent variables to represent problem situations? Problems that occur in business situations often require expressing income as a linear function of one variable like time worked or number of sales. For example, if an employee earns $7.25 per hour, then

More information

Functions: Polynomial, Rational, Exponential

Functions: Polynomial, Rational, Exponential Functions: Polynomial, Rational, Exponential MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Spring 2014 Objectives In this lesson we will learn to: identify polynomial expressions,

More information

2. What is the x-intercept of line B? (a) (0, 3/2); (b) (0, 3); (c) ( 3/2, 0); (d) ( 3, 0); (e) None of these.

2. What is the x-intercept of line B? (a) (0, 3/2); (b) (0, 3); (c) ( 3/2, 0); (d) ( 3, 0); (e) None of these. Review Session, May 19 For problems 1 4 consider the following linear equations: Line A: 3x y = 7 Line B: x + 2y = 3 1. What is the y-intercept of line A? (a) ( 7/, 0); (b) (0, 7/); (c) (0, 7); (d) (7,

More information

The plot shows the graph of the function f (x). Determine the quantities.

The plot shows the graph of the function f (x). Determine the quantities. MATH 211 SAMPLE EXAM 1 SOLUTIONS 6 4 2-2 2 4-2 1. The plot shows the graph of the function f (x). Determine the quantities. lim f (x) (a) x 3 + Solution: Look at the graph. Let x approach 3 from the right.

More information

Section 1.4. Meaning of Slope for Equations, Graphs, and Tables

Section 1.4. Meaning of Slope for Equations, Graphs, and Tables Section 1.4 Meaning of Slope for Equations, Graphs, and Tables Finding Slope from a Linear Equation Finding Slope from a Linear Equation Example Find the slope of the line Solution Create a table using

More information

2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course.

2014 Summer Review for Students Entering Algebra 2. TI-84 Plus Graphing Calculator is required for this course. 1. Solving Linear Equations 2. Solving Linear Systems of Equations 3. Multiplying Polynomials and Solving Quadratics 4. Writing the Equation of a Line 5. Laws of Exponents and Scientific Notation 6. Solving

More information

Middle School Math Course 3

Middle School Math Course 3 Middle School Math Course 3 This course covers the topics shown below. Students navigate learning paths based on their level of readiness. Institutional users may customize the scope and sequence to meet

More information

Chapter 2 Polynomial and Rational Functions

Chapter 2 Polynomial and Rational Functions SECTION.1 Linear and Quadratic Functions Chapter Polynomial and Rational Functions Section.1: Linear and Quadratic Functions Linear Functions Quadratic Functions Linear Functions Definition of a Linear

More information

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY

2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY 2017 SUMMER REVIEW FOR STUDENTS ENTERING GEOMETRY The following are topics that you will use in Geometry and should be retained throughout the summer. Please use this practice to review the topics you

More information

Chapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter P. Prerequisites. Slide P- 1. Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 1 Chapter P Prerequisites 1 P.1 Real Numbers Quick Review 1. List the positive integers between -4 and 4.. List all negative integers greater than -4. 3. Use a calculator to evaluate the expression

More information

MATHEMATICAL METHODS UNIT 1 Chapter 1 Reviewing Linear Equations Chapter 2 Coordinate geometry & linear relations

MATHEMATICAL METHODS UNIT 1 Chapter 1 Reviewing Linear Equations Chapter 2 Coordinate geometry & linear relations REVIEWING LINEAR EQUATIONS E da = q ε ( B da = 0 E ds = dφ. B ds = μ ( i + μ ( ε ( dφ 3 MATHEMATICAL METHODS UNIT 1 Chapter 1 Reviewing Linear Equations Chapter 2 Coordinate geometry & linear relations

More information

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions

Beginning Algebra. 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions 1. Review of Pre-Algebra 1.1 Review of Integers 1.2 Review of Fractions Beginning Algebra 1.3 Review of Decimal Numbers and Square Roots 1.4 Review of Percents 1.5 Real Number System 1.6 Translations:

More information

Minnesota K-12 Academic Standards in Mathematics (2007)

Minnesota K-12 Academic Standards in Mathematics (2007) 8.1.1.1 Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number an irrational

More information

Chapter 2: Rational. Functions. SHMth1: General Mathematics. Accountancy, Business and Management (ABM. Mr. Migo M. Mendoza

Chapter 2: Rational. Functions. SHMth1: General Mathematics. Accountancy, Business and Management (ABM. Mr. Migo M. Mendoza Chapter 2: Rational Functions SHMth1: General Mathematics Accountancy, Business and Management (ABM Mr. Migo M. Mendoza Chapter 2: Rational Functions Lecture 6: Basic Concepts Lecture 7: Solving Rational

More information

3. Find the slope of the tangent line to the curve given by 3x y e x+y = 1 + ln x at (1, 1).

3. Find the slope of the tangent line to the curve given by 3x y e x+y = 1 + ln x at (1, 1). 1. Find the derivative of each of the following: (a) f(x) = 3 2x 1 (b) f(x) = log 4 (x 2 x) 2. Find the slope of the tangent line to f(x) = ln 2 ln x at x = e. 3. Find the slope of the tangent line to

More information

Simplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4)

Simplify each numerical expression. Show all work! Only use a calculator to check. 1) x ) 25 ( x 2 3) 3) 4) NAME HONORS ALGEBRA II REVIEW PACKET To maintain a high quality program, students entering Honors Algebra II are expected to remember the basics of the mathematics taught in their Algebra I course. In

More information

Equations and Inequalities

Equations and Inequalities Equations and Inequalities Figure 1 CHAPTER OUTLINE 1 The Rectangular Coordinate Systems and Graphs Linear Equations in One Variable Models and Applications Comple Numbers Quadratic Equations 6 Other Types

More information

Chapter 5: Writing Linear Equations Study Guide (REG)

Chapter 5: Writing Linear Equations Study Guide (REG) Chapter 5: Writing Linear Equations Study Guide (REG) 5.1: Write equations of lines given slope and y intercept or two points Write the equation of the line with the given information: Ex: Slope: 0, y

More information

Introduction to systems of equations

Introduction to systems of equations Introduction to systems of equations A system of equations is a collection of two or more equations that contains the same variables. This is a system of two equations with two variables: In solving a

More information

Chapter 1 Review Applied Calculus 31

Chapter 1 Review Applied Calculus 31 Chapter Review Applied Calculus Section : Linear Functions As you hop into a taxicab in Allentown, the meter will immediately read $.0; this is the drop charge made when the taximeter is activated. After

More information

Copyright 2015 Edmentum All rights reserved.

Copyright 2015 Edmentum All rights reserved. Copyright 2015 Edmentum All rights reserved. Linear Equations & Graphs 1. A line has a y intercept of and a slope of. Find the equation of the line. A. B. C. D. Evaluate Functions 2. The graph of the function

More information

Course Readiness and Skills Review Handbook (Topics 1-10, 17) (240 topics, due. on 09/11/2015) Course Readiness (55 topics)

Course Readiness and Skills Review Handbook (Topics 1-10, 17) (240 topics, due. on 09/11/2015) Course Readiness (55 topics) Course Name: Gr. 8 Fall 2015 Course Code: C6HNH-TEK9E ALEKS Course: Middle School Math Course 3 Instructor: Mr. Fernando Course Dates: Begin: 08/31/2015 End: 06/17/2016 Course Content: 642 Topics (637

More information

Systems of Linear Equations

Systems of Linear Equations Systems of Linear Equations As stated in Section G, Definition., a linear equation in two variables is an equation of the form AAAA + BBBB = CC, where AA and BB are not both zero. Such an equation has

More information

TEST 150 points

TEST 150 points Math 130 Spring 008 Name: TEST #1 @ 150 points Write neatly. Show all work. Write all responses on separate paper. Clearly label the exercises. 1. A piecewise-defined function is given. 1- x if x< f (

More information

MTH103 Section 065 Exam 2. x 2 + 6x + 7 = 2. x 2 + 6x + 5 = 0 (x + 1)(x + 5) = 0

MTH103 Section 065 Exam 2. x 2 + 6x + 7 = 2. x 2 + 6x + 5 = 0 (x + 1)(x + 5) = 0 Absolute Value 1. (10 points) Find all solutions to the following equation: x 2 + 6x + 7 = 2 Solution: You first split this into two equations: x 2 + 6x + 7 = 2 and x 2 + 6x + 7 = 2, and solve each separately.

More information

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0).

This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This is Solving Linear Systems, chapter 4 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/

More information

AP Calculus I Summer Packet

AP Calculus I Summer Packet AP Calculus I Summer Packet This will be your first grade of AP Calculus and due on the first day of class. Please turn in ALL of your work and the attached completed answer sheet. I. Intercepts The -intercept

More information

College Algebra Through Problem Solving (2018 Edition)

College Algebra Through Problem Solving (2018 Edition) City University of New York (CUNY) CUNY Academic Works Open Educational Resources Queensborough Community College Winter 1-25-2018 College Algebra Through Problem Solving (2018 Edition) Danielle Cifone

More information

Solving Linear and Rational Inequalities Algebraically. Definition 22.1 Two inequalities are equivalent if they have the same solution set.

Solving Linear and Rational Inequalities Algebraically. Definition 22.1 Two inequalities are equivalent if they have the same solution set. Inequalities Concepts: Equivalent Inequalities Solving Linear and Rational Inequalities Algebraically Approximating Solutions to Inequalities Graphically (Section 4.4).1 Equivalent Inequalities Definition.1

More information

Glossary. Glossary 981. Hawkes Learning Systems. All rights reserved.

Glossary. Glossary 981. Hawkes Learning Systems. All rights reserved. A Glossary Absolute value The distance a number is from 0 on a number line Acute angle An angle whose measure is between 0 and 90 Addends The numbers being added in an addition problem Addition principle

More information

Geometry 21 Summer Work Packet Review and Study Guide

Geometry 21 Summer Work Packet Review and Study Guide Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the

More information

Functions of Several Variables: Limits and Continuity

Functions of Several Variables: Limits and Continuity Functions of Several Variables: Limits and Continuity Philippe B. Laval KSU Today Philippe B. Laval (KSU) Limits and Continuity Today 1 / 24 Introduction We extend the notion of its studied in Calculus

More information

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i

Final Exam C Name i D) 2. Solve the equation by factoring. 4) x2 = x + 72 A) {1, 72} B) {-8, 9} C) {-8, -9} D) {8, 9} 9 ± i Final Exam C Name First, write the value(s) that make the denominator(s) zero. Then solve the equation. 7 ) x + + 3 x - = 6 (x + )(x - ) ) A) No restrictions; {} B) x -, ; C) x -; {} D) x -, ; {2} Add

More information