Systems of Linear Equations
|
|
- Alvin Spencer
- 5 years ago
- Views:
Transcription
1 4 Systems of Linear Equations Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 1 1-1
2 4.3 Applications of R.1 Systems Fractions of Linear Equations Objectives 1. Solve geometric problems using two variables. 2. Solve money problems using two variables. 3. Solve mixture problems using two variables. 4. Solve distance-rate problems using two variables. 5. Solve problems with three variables using a system of three equations. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 2 1-2
3 Solving an Applied Problem by Writing a System of Equations Step 1 Read the problem, several times if necessary. What information is given? What is to be found? This is often stated in the last sentence. Step 2 Assign variables to represent the unknown values. Use a sketch, diagram, or table, as needed. Step 3 Write a system of equations using the variable expressions. Step 4 Solve the system of equations. Step 5 State the answer to the problem. Label it appropriately. Does it seem reasonable? Step 6 Check the answer in the words of the original problem. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 3 1-3
4 Example 1 Finding the Dimensions of a Parking Lot The length of a rectangular parking lot is 20 ft more than three times its width. The perimeter of the parking lot is 800 ft. What are the dimensions of the parking lot? Step 1 Read the problem again. We must find the dimensions of the parking lot. Step 2 Assign variables. Let L = the length and W = the width. L W Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 4 1-4
5 Finding the Dimensions of a Parking Lot The length of a rectangular parking lot is 20 ft more than three times its width. The perimeter of the parking lot is 800 ft. What are the dimensions of the parking lot? Step 3 Write a system of equations. Because the perimeter is 800 ft, we find one equation by using the perimeter formula: 2L + 2W = 800. Because the length is 20 ft more than three times its width, we have The system is, therefore, L = 3W L + 2W = 800 (1) L = 3W (2) Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 5 1-5
6 EXAMPLE Continued. 1 Finding the Dimensions of a Parking Lot The length of a rectangular parking lot is 20 ft more than three times its width. The perimeter of the parking lot is 800 ft. What are the dimensions of the parking lot? Step 4 Solve the system of equations. We substitute 3W + 20 for L, in equation (1), and solve for W. 2L + 2W = 800 (1) 2(3W + 20) + 2W = 800 Let L = 3W W W = 800 8W + 40 = 800 Distributive property Combine terms. 8W = 760 Subtract 40. W = 95 Divide by 8. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 6 1-6
7 Finding the Dimensions of a Parking Lot The length of a rectangular parking lot is 20 ft more than three times its width. The perimeter of the parking lot is 800 ft. What are the dimensions of the parking lot? Step 4 Solve the system of equations. We just solved the equation 2L + 2W = 800 and found W = 95. Now, let W = 95 in the equation L = 3W + 20 to find L. L = 3W + 20 L = 3(95) + 20 Let W = 95. L = 305 Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 7 1-7
8 EXAMPLE Continued. 1 Finding the Dimensions of a Parking Lot The length of a rectangular parking lot is 20 ft more than three times its width. The perimeter of the parking lot is 800 ft. What are the dimensions of the parking lot? Step 5 State the answer. The length is 305 ft and the width is 95 ft. Step 6 Check. The perimeter is 2(305) + 2(95) = 800 ft, and the length, 305 ft, is 20 ft more than three times the width, since 3(95) + 20 = 305. The answer is correct. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 8 1-8
9 Example 2 Solving a Problem about Prices At a local restaurant, the price of 1 soft drink and 2 hamburgers is $12.15 and EXAMPLE 2 the price of 5 soft drinks and 4 hamburgers is $ Find the price of a single hamburger and a soft drink. Step 1 Read the problem again. There are two unknowns. Step 2 Assign variables. Let s represent the price of one soft drink and h represent the price of one hamburger. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide 9 1-9
10 Solving a Problem about Prices At a local restaurant, the price of 1 soft drink and 2 hamburgers is $12.15 and EXAMPLE 2 the price of 5 soft drinks and 4 hamburgers is $ Find the price of a single hamburger and a soft drink. Step 3 Write a system of equations. Because one soft drink and 2 hamburgers cost a total of $12.15, one equation for the system is s + 2h = By similar reasoning, the second equation is Therefore, the system is 5s + 4h = s + 2h = (1) 5s + 4h = (2) Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
11 Solving a Problem about Prices At a local restaurant, the price of 1 soft drink and 2 hamburgers is $12.15 and EXAMPLE 2 the price of 5 soft drinks and 4 hamburgers is $ Find the price of a single hamburger and a soft drink. Step 4 Solve the system of equations. s + 2h = (1) 5s + 4h = (2) 5s 10h = Multiply each side of (1) by 5. 5s + 4h = (2) 6h = Add. h = 4.75 Divide by 6. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
12 Solving a Problem about Prices At a local restaurant, the price of 1 soft drink and 2 hamburgers is $12.15 and EXAMPLE 2 the price of 5 soft drinks and 4 hamburgers is $ Find the price of a single hamburger and a soft drink. Step 4 Solve the system of equations. We just found h = Now, let h = 4.75 in the equation s + 2h = to find s. s + 2h = (1) s + 2(4.75) = Let h = s = Multiply. s = 2.65 Subtract Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
13 Solving a Problem about Prices At a local restaurant, the price of 1 soft drink and 2 hamburgers is $12.15 and EXAMPLE 2 the price of 5 soft drinks and 4 hamburgers is $ Find the price of a single hamburger and a soft drink. Step 5 State the answer. The price of a single soft drink is $2.65 and the price of a hamburger is $4.75. Step 6 Check that these values satisfy the conditions stated in the problem. 5(2.65) 1(2.65) + 4(4.75) 2(4.75) = $32.25 $12.15 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
14 Example 3 Solving a Mixture Problem How many ounces each of 10% hydrochloric acid and 25% hydrochloric acid must be combined to get 40 oz of solution that is 22% hydrochloric acid? Step 1 Read the problem. Two solutions of different strengths are being mixed together to get a specific amount of a solution with an inbetween strength. Step 2 Assign a variable. Let x = the number of ounces of 10% solution and y = the number of ounces of 25% solution. 25% + = 10% 25% 22% 10% x oz y oz 40 oz Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
15 Solving a Mixture Problem How many ounces each of 10% hydrochloric acid and 25% hydrochloric acid must be combined to get 40 oz of solution that is 22% hydrochloric acid? Step 2 Assign a variable. Let x = the number of ounces of 10% solution and y = the number of ounces of 25% solution. Percent (as a decimal) 10% = % = % = 0.22 Number of Ounces x y 40 Ounces of Pure Acid 0.10x 0.25y 0.22(40) Step 3 Write a system of equations. + x + y = = 40 10% 25% 22%.10x x oz y oz +.25y = oz (1) (2) Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
16 Solving a Mixture Problem How many ounces each of 10% hydrochloric acid and 25% hydrochloric acid must be combined to get 40 oz of solution that is 22% hydrochloric acid? Step 4 Solve the system. x y 40 + = (1) 0.10x 0.25y = (2) 10x 10y = 10x 25y = Multiply each side of (1) by 10. Multiply each side of (2) by y = 480 Add. y = 32 Divide by 15. Because y = 32 and x + y = 40, x = 8. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
17 Solving a Mixture Problem How many ounces each of 10% hydrochloric acid and 25% hydrochloric acid must be combined to get 40 oz of solution that is 22% hydrochloric acid? Step 5 State the answer. The desired mixture will require 8 oz of the 10% solution and 32 oz of the 25% solution. Step 6 Check that these values satisfy both equations of the system. Percent (as a decimal) Number of Ounces 8 32 Ounces of Pure Acid Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
18 Example 4 Solving a Motion Problem A car travels at 310 miles in the same time that a bus travels 290 miles. If the speed of the car is 4 mph faster than the speed of the bus, find both speeds. Step 1 Read the problem again. Given the distances traveled, we need to find the speed of each vehicle. Step 2 Assign variables. Let x = the speed of the car Since d = rt, and y = the speed of the bus. t = d r. distance (d) rate (r) time (t) Car 310 x 310 x Bus 290 y 290 y Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
19 Solving a Motion Problem A car travels at 310 miles in the same time that a bus travels 290 miles. If the speed of the car is 4 mph faster than the speed of the bus, find both speeds. Step 3 Write a system of equations. The problem states that the car travels 4 mph faster than the bus. Since the two speeds are x and y, x = y + 4. Both vehicles travel for the same time, so from the table x = y. Multiplying both sides by xy gives 310y = 290x. distance (d) rate (r) time (t) Car 310 x 310 x Bus 290 y 290 y Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
20 Solving a Motion Problem A car travels at 310 miles in the same time that a bus travels 290 miles. If the speed of the car is 4 mph faster than the speed of the bus, find both speeds. Step 4 Solve the system of equations using substitution. x = y + 4 (1) 310y = 290x (2) 310y = 290x (2) 310y = 290(y + 4) Let x = y y = 290y Distributive property 20y = 1160 Subtract 290y. y = 58 Divide by 20. Because x = y + 4, the value of x is = 62. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
21 Solving a Motion Problem A car travels at 310 miles in the same time that a bus travels 290 miles. If the speed of the car is 4 mph faster than the speed of the bus, find both speeds. Step 5 State the answer. The car s speed is 62 mph, and the speed of the bus is 58 mph. Step 6 Check. This is especially important since one of the equations had variable denominators. Car: t = d r = Bus: t = d r = = 5 = 5 Times are equal. Since = 4, the conditions of the problem are satisfied. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
22 Example 6 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 1 Read the problem again. There are three unknowns. Step 2 Assign variables to represent the three unknowns. Let x = number of loaves of honey wheat, y = number of loaves of sunflower, and z = number of loaves of French bread. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
23 Example 6 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 3 Write a system of equations. x = loaves of honey wheat, y = loaves of sunflower, z = loaves of French bread y = 2x, or y 2x = 0 (1) z = y 3, or z y = 2 (2) 2.69x y z = (3) Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
24 EXAMPLE Continued. 5 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 4 Solve the system of three equations using substitution. y = 2x (1) z = y 3 (2) z = 2x 3 Let y = 2x. (4) Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
25 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 4 Solve the system of three equations using substitution. y = 2x z = 2x 3 (1) (4) 269x + 289y + 339z = 9658 Multiply each side (3) by x + 289(2x) + 339(2x 3) = 9658 Substitute in (3). Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
26 EXAMPLE Continued. 5 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 4 Solve the system of three equations using substitution. 269x + 578x + 678x 1017 = x 1017 = 9658 Multiply & distribute. Combine terms. 1525x = 10,675 Add x = 7 Divide by Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
27 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 4 Solve the system of three equations using substitution. Because x = 7 and y = 2x, y = 14. Also, because y = 14 and z = y 3, z = 11. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
28 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 5 State the answer. The solution set is { (7, 14, 11) }, meaning that 7 loaves of honey wheat, 14 loaves of sunflower, and 11 loaves of French bread were sold. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
29 EXAMPLE 5 Solving a Problem Involving Prices At a local bakery, a loaf of honey wheat bread costs $2.69, a loaf of sunflower bread costs $2.89, and a loaf of French bread costs $3.39. On a recent day, twice as many loaves of sunflower were sold as honey wheat. The number of loaves of French bread sold was three less than the number of loaves of sunflower. Total receipts for these breads were $ How many loaves of each type of bread were sold? Step 6 Check. Since 14 = 2(7), the number of loaves of sunflower sold is twice the number of loaves as honey wheat. Also, 14 3 = 11, so the number of loaves of French bread is three less than the number of loaves of sunflower. The total from the receipts is $96.58 as stated. 2.69(7) (14) (11) = Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.3, Slide
Chapters 4/5 Class Notes. Intermediate Algebra, MAT1033C. SI Leader Joe Brownlee. Palm Beach State College
Chapters 4/5 Class Notes Intermediate Algebra, MAT1033C Palm Beach State College Class Notes 4.1 Professor Burkett 4.1 Systems of Linear Equations in Two Variables A system of equations is a set of two
More information2.2. Formulas and Percent. Objectives. Solve a formula for a specified variable. Solve applied problems by using formulas. Solve percent problems.
Chapter 2 Section 2 2.2 Formulas and Percent Objectives 1 2 3 4 Solve a formula for a specified variable. Solve applied problems by using formulas. Solve percent problems. Solve problems involving percent
More informationName Class Date. Expanding Algebraic Expressions. 4. Use the Distributive Property to expand the expression 7(9x 2).
Name Class Date Practice 7-1 Expanding Algebraic Expressions 7-1 Expanding Algebraic Expressions 1. Find a sum equivalent to the product 6(y + x). 2. Find a difference equivalent to the product 11(x y).
More information8.4. Systems of Equations in Three Variables. Identifying Solutions 2/20/2018. Example. Identifying Solutions. Solving Systems in Three Variables
8.4 Systems of Equations in Three Variables Copyright 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Identifying Solutions Solving Systems in Three Variables Dependency, Inconsistency,
More informationTranslate from words to mathematical expressions. Distinguish between simplifying expressions and solving equations.
Chapter 2 Section 3 2.3 Applications of Linear Equations Objectives 1 2 3 4 5 6 7 Translate from words to mathematical expressions. Write equations from given information. Distinguish between simplifying
More informationChapter 6. Additional Topics in Trigonometry. 6.6 Vectors. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 6 Additional Topics in Trigonometry 6.6 Vectors Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Obectives: Use magnitude and direction to show vectors are equal. Visualize scalar multiplication,
More informationChapter 7 Linear Systems and Matrices
Chapter 7 Linear Systems and Matrices Overview: 7.1 Solving Systems of Equations 7.2 Systems of Linear Equations in Two Variables 7.3 Multivariable Linear Systems 7.1 Solving Systems of Equations What
More informationPercent Change of Dimensions
Percent Change of Dimensions Reteaching 71 Math Course 3, Lesson 71 Dilation: Add the percent of increase to 100%. Reduction: Subtract the percent of decrease from 100%. Scale factor: To find the scale
More informationSolve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x. 2) 7x - (3x - 1) = 2. 3) 2x 5 - x 3 = 2 4) 15. 5) -4.2q =
Spring 2011 Name Math 115 Elementary Algebra Review Wednesday, June 1, 2011 All problems must me done on 8.5" x 11" lined paper. Solve. Label any contradictions or identities. 1) -4x + 2(3x - 3) = 5-9x
More informationSystems of Equations and Applications
456 CHAPTER 8 Graphs, Functions, and Systems of Equations and Inequalities 8.7 Systems of Equations and Applications Linear Systems in Two Variables The worldwide personal computer market share for different
More informationWrite an equation for each relationship. Then make a table of input-output pairs and tell whether the function is proportional.
Functions Reteaching 41 Math Course, Lesson 41 A function is a rule that identifies a relationship between a set of input numbers and a set of output numbers. A function rule can be described in words,
More informationSystems of Linear Equations
4 Systems of Linear Equations Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 4.1, Slide 1 1-1 4.1 Systems of Linear Equations in Two Variables R.1 Fractions Objectives 1. Decide whether an
More informationChapter 8 RADICAL EXPRESSIONS AND EQUATIONS
Name: Instructor: Date: Section: Chapter 8 RADICAL EXPRESSIONS AND EQUATIONS 8.1 Introduction to Radical Expressions Learning Objectives a Find the principal square roots and their opposites of the whole
More informationSection 2.1 Objective 1: Determine If a Number Is a Solution of an Equation Video Length 5:19. Definition A in is an equation that can be
Section 2.1 Video Guide Linear Equations: The Addition and Multiplication Properties of Equality Objectives: 1. Determine If a Number Is a Solution of an Equation 2. Use the Addition Property of Equality
More informationMATH98 Intermediate Algebra Practice Test Form A
MATH98 Intermediate Algebra Practice Test Form A MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the equation. 1) (y - 4) - (y + ) = 3y 1) A)
More informationSection 3.2 Applications of Radian Measure
Section. Applications of Radian Measure 07 (continued) 88. 80 radians = = 0 5 5 80 radians = = 00 7 5 = 5 radian s 0 = 0 radian s = radian 80 89. (a) In hours, the hour hand will rotate twice around the
More information2.1 Solving Equations Using Properties of Equality Math 085 Chapter 2. Chapter 2
2.1 Solving Equations Using Properties of Equality Math 085 Chapter 2 Chapter 2 2.1 Solving Equations Using Properties of Equality 2.2 More about Solving Equations 2.3 Application of Percent 2.4 Formulas
More informationChapter 6 RATIONAL EXPRESSIONS AND EQUATIONS
Name: Instructor: Date: Section: Chapter 6 RATIONAL EXPRESSIONS AND EQUATIONS 6.1 Multiplying and Simplifying Rational Expressions Learning Objectives a Find all numbers for which a rational expression
More informationMini-Lecture 2.1 Simplifying Algebraic Expressions
Copyright 01 Pearson Education, Inc. Mini-Lecture.1 Simplifying Algebraic Expressions 1. Identify terms, like terms, and unlike terms.. Combine like terms.. Use the distributive property to remove parentheses.
More informationTranslate from words to mathematical expressions.
2.3 Applications of Linear Equations Objectives 1 2 Write equations from given information. There are usually key ords and phrases in a verbal problem that translate into mathematical expressions involving
More informationFundamentals. Copyright Cengage Learning. All rights reserved.
Fundamentals Copyright Cengage Learning. All rights reserved. 1.6 Modeling with Equations Copyright Cengage Learning. All rights reserved. Objectives Making and Using Models Problems About Interest Problems
More information5.1. Integer Exponents and Scientific Notation. Objectives. Use the product rule for exponents. Define 0 and negative exponents.
Chapter 5 Section 5. Integer Exponents and Scientific Notation Objectives 2 4 5 6 Use the product rule for exponents. Define 0 and negative exponents. Use the quotient rule for exponents. Use the power
More informationLet s Be Rational Practice Answers
Investigation Additional Practice. a. fractions:,,,,, ;,,,,, ;,,,,,,,, ;,,,,, ;,,,,, ;,, ;, ;,, ;,,, ;,, ;,, c.,,. a., because each fraction is,., because each fraction is #. c., because each fraction
More informationIntroductory Algebra Chapter 9 Review
Introductory Algebra Chapter 9 Review Objective [9.1a] Find the principal square roots and their opposites of the whole numbers from 0 2 to 2 2. The principal square root of a number n, denoted n,is the
More informationFinal Exam Review for DMAT 0310
Final Exam Review for DMAT 010 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Factor the polynomial completely. What is one of the factors? 1) x
More informationGeometry - Summer 2016
Geometry - Summer 2016 Introduction PLEASE READ! The purpose of providing summer work is to keep your skills fresh and strengthen your base knowledge so we can build on that foundation in Geometry. All
More information= = =
. D - To evaluate the expression, we can regroup the numbers and the powers of ten, multiply, and adjust the decimal and exponent to put the answer in correct scientific notation format: 5 0 0 7 = 5 0
More informationMath 101, Basic Algebra. Solving Linear Equations and Inequalities
Math 101, Basic Algebra Author: Debra Griffin Name Chapter 2 Solving Linear Equations and Inequalities 2.1 Simplifying Algebraic Expressions 2 Terms, coefficients, like terms, combining like terms, simplifying
More informationKwan went to the store with $20 and left the store with his purchases and $7.35. How much money did Kwan spend?
Name Score Benchmark Test 1 Math Course 2 For use after Lesson 0 1. (1) At Washington School there are 2 classrooms and an average of 25 students in each classroom. Which equation shows how to find the
More informationExponents, Polynomials, and Polynomial Functions. Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1
5 Exponents, Polynomials, and Polynomial Functions Copyright 2014, 2010, 2006 Pearson Education, Inc. Section 5.1, 1 5.1 Integer Exponents R.1 Fractions and Scientific Notation Objectives 1. Use the product
More informationUnit 3 Exam Review Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Unit Eam Review Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Some Useful Formulas: Compound interest formula: A=P + r nt n Continuously
More informationChapter 02 Test. 1 Solve the proportion. A 60 B 60 C 1.35 D Solve the proportion. A 5.4 B 7.4 C 1959 D 1.625
Name: ate: 1 Solve the proportion. 60 60 1.35 1.35 2 Solve the proportion. 5.4 7.4 1959 1.625 3 Solve the equation. 8x 5 = 12x + 1 1 opyright 2005-2006 by Pearson Education Page 1 of 7 4 Solve the equation.
More informationMath Mammoth End-of-the-Year Test, Grade 6, Answer Key
Math Mammoth End-of-the-Year Test, Grade 6, Answer Key Please see the test for grading instructions. The Basic Operations 1. a. 2,000 38 = 52 R4. There will be 52 bags of cinnamon. 2. a. 2 5 = 32 b. 5
More informationAlgebra 2 Honors Summer Review
Algebra Honors Summer Review 07-08 Label each problem and do all work on separate paper. All steps in your work must be shown in order to receive credit. No Calculators Allowed. Topic : Fractions A. Perform
More informationMath 074 Final Exam Review. REVIEW FOR NO CALCULATOR PART OF THE EXAM (Questions 1-14)
Math 074 Final Exam Review REVIEW FOR NO CALCULATOR PART OF THE EXAM (Questions -4) I. Can you add, subtract, multiply and divide fractions and mixed numbers?. Perform the indicated operations. Be sure
More informationRemember, you may not use a calculator when you take the assessment test.
Elementary Algebra problems you can use for practice. Remember, you may not use a calculator when you take the assessment test. Use these problems to help you get up to speed. Perform the indicated operation.
More information2x + 5 = 17 2x = 17 5
1. (i) 9 1 B1 (ii) 19 1 B1 (iii) 7 1 B1. 17 5 = 1 1 = x + 5 = 17 x = 17 5 6 3 M1 17 (= 8.5) or 17 5 (= 1) M1 for correct order of operations 5 then Alternative M1 for forming the equation x + 5 = 17 M1
More information13. Convert to a mixed number: Convert to an improper fraction: Are these two fractions equivalent? 7
FINAL REVIEW WORKSHEET BASIC MATH Chapter 1. 1. Give the place value of 7 in 3, 738, 500. 2. Give the word name for 302, 525. 3. Write two million, four hundred thirty thousand as a numeral. 4. Name the
More information1. f(x) = f(x) = 3. y 2-3y p - 4 8p2. Math 0312 EXAM 3 Review Questions. Name. Find all numbers not in the domain of the function.
Name Find all numbers not in the domain of the function. 1. f(x) = 8 x - 5 Find all numbers that are not in the domain of the function. Then give the domain using set notation. 10 2. f(x) = x 2 + 11x +
More informationAssignment: Summer Assignment Part 1 of 8 Real Numbers and Their Properties. Student: Date:
Student: Date: Assignment: Summer Assignment Part of 8 Real Numbers and Their Properties. Identify to which number groups (natural numbers, whole numbers, integers, rational numbers, real numbers, and
More informationCurriculum Catalog
2016-2017 Curriculum Catalog 2016 Glynlyon, Inc. Table of Contents ALGEBRA I FUNDAMENTALS COURSE OVERVIEW... ERROR! BOOKMARK NOT DEFINED. UNIT 1: FOUNDATIONS OF ALGEBRA... ERROR! BOOKMARK NOT DEFINED.
More informationName. Unit 1 Worksheets Math 150 College Algebra and Trig
Name Unit 1 Worksheets Math 10 College Algebra and Trig Revised: Fall 009 Worksheet 1: Integral Eponents Simplify each epression. Write all answers in eponential form. 1. (8 ). ( y). (a b ). y 6. (7 8
More informationApplications of Rational Expressions
6.5 Applications of Rational Expressions 1. Find the value of an unknown variable in a formula. 2. Solve a formula for a specified variable. 3. Solve applications using proportions. 4. Solve applications
More informationChapter 5 Newton s Laws of Motion. Copyright 2010 Pearson Education, Inc.
Chapter 5 Newton s Laws of Motion Copyright 2010 Pearson Education, Inc. Force and Mass Copyright 2010 Pearson Education, Inc. Units of Chapter 5 Newton s First Law of Motion Newton s Second Law of Motion
More information4. Based on the table below, what is the joint relative frequency of the people surveyed who do not have a job and have a savings account?
Name: Period: Date: Algebra 1 Common Semester 1 Final Review Like PS4 1. How many surveyed do not like PS4 and do not like X-Box? 2. What percent of people surveyed like the X-Box, but not the PS4? 3.
More informationa. Bob: 7, Bridget: 4, Brian 1 b. Bob: 7, Bridget: 4, Brian 3 c. Bob: 7, Bridget: 14, Brian 3 a. 100 b. 150 c c. 2 d.
Period: Date: K. Williams 8th Grade Year Review: Chapters -4. A neighborhood pool charges $22 for a pool membership plus an additional $2 for each visit to the pool. If Elliot visited the pool 6 times,
More information7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems. Main Ideas:
7.2 Solving Systems with Graphs Name: Date: Goal: to use the graphs of linear equations to solve linear systems Toolkit: graphing lines rearranging equations substitution Main Ideas: Definitions: Linear
More informationCurriculum Catalog
2017-2018 Curriculum Catalog 2017 Glynlyon, Inc. Table of Contents ALGEBRA I COURSE OVERVIEW... 1 UNIT 1: FOUNDATIONS OF ALGEBRA... 1 UNIT 2: LINEAR EQUATIONS... 2 UNIT 3: FUNCTIONS... 2 UNIT 4: INEQUALITIES...
More informationMath 100 Final Exam Review
Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.
More informationMath 100 Final Exam Review
Math 0 Final Eam Review Name The problems included in this review involve the important concepts covered this semester. Work in groups of 4. If our group gets stuck on a problem, let our instructor know.
More informationHonors and Regular Algebra II & Trigonometry Summer Packet
Honors and Regular Algebra II & Trigonometry Summer Packet Hello Students, Parents, and Guardians! I hope everyone is enjoying the summer months and the time to rela and recoup. To add to your summer fun,
More informationSummer Packet for Students entering Algebra 1/2
Course 3/Pre-Algebra (formerly: Entering Algebra /2): Page of 6 Name Date Period Directions: Please show all work on a separate sheet of paper and place your final answer on your summer packet. If no work
More informationMATH 081. Diagnostic Review Materials PART 2. Chapters 5 to 7 YOU WILL NOT BE GIVEN A DIAGNOSTIC TEST UNTIL THIS MATERIAL IS RETURNED.
MATH 08 Diagnostic Review Materials PART Chapters 5 to 7 YOU WILL NOT BE GIVEN A DIAGNOSTIC TEST UNTIL THIS MATERIAL IS RETURNED DO NOT WRITE IN THIS MATERIAL Revised Winter 0 PRACTICE TEST: Complete as
More information4) A high school graduating class is made up of 550 students. There are 144 more boys than girls. How many girls are in the class?
Math 110 FYC chapter and 4 Practice The actual text is different Solve. 1) The difference of four times a number and seven times the same number is 9. Find the number. 1) 2) x - 1 = 1 15 2) ) 5 4 x = 15
More informationNegative Exponents Scientific Notation for Small Numbers
Negative Exponents Scientific Notation for Small Numbers Reteaching 51 Math Course 3, Lesson 51 The Law of Exponents for Negative Exponents An exponential expression with a negative exponent is the reciprocal
More informationHuron School District Core Curriculum Guide Grade Level: 4th Content Area: Math
Unit Title: Understand Whole Numbers and Operations Month(s): August, September, October 4N3.1; 4N1.1; 4A3.1; 4A1.3 4A1.2; 4A2.1; 4A2.2; 4A4.1 4A1.1 To read, write, and indentify the place value of whole
More informationChapter 1: Fundamentals of Algebra Lecture notes Math 1010
Section 1.1: The Real Number System Definition of set and subset A set is a collection of objects and its objects are called members. If all the members of a set A are also members of a set B, then A is
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 1: Interpreting Structure in Expressions Instruction
Prerequisite Skills This lesson requires the use of the following skills: evaluating expressions using the order of operations evaluating expressions for a given value identifying parts of an expression
More information4. Based on the table below, what is the joint relative frequency of the people surveyed who do not have a job and have a savings account?
Name: Period: Date: Algebra 1 Common Semester 1 Final Review 1. How many surveyed do not like PS4 and do not like X-Box? 2. What percent of people surveyed like the X-Box, but not the PS4? 3. What is the
More informationProject - Math 99 Final Practice Due Thursday 3 rd August
Project - Math 99 Final Practice Summer 017 Project - Math 99 Final Practice Due Thursday rd August Section 1: Multiple Choice Questions Students Name : 1. The epression 10 + when factored fully is:- A.
More informationASU Mathematics Placement Test Sample Problems June, 2000
ASU Mathematics Placement Test Sample Problems June, 000. Evaluate (.5)(0.06). Evaluate (.06) (0.08). Evaluate ( ) 5. Evaluate [ 8 + ( 9) ] 5. Evaluate 7 + ( ) 6. Evaluate ( 8) 7. Evaluate 5 ( 8. Evaluate
More informationChapter 1. Expressions, Equations, and Functions
Chapter 1 Expressions, Equations, and Functions 1.1 Evaluate Expressions I can evaluate algebraic expressions and use exponents. CC.9-12.N.Q.1 Vocabulary: Variable a letter used to represent one or more
More information3.1 Algebraic Expressions. Parts of an algebraic expression are called terms. One way to simplify an expression is to combine like terms. 4x 2.
3.1 Algebraic Expressions Parts of an algebraic expression are called terms. One way to simplify an expression is to combine like terms. What does it mean to combine like terms? 4x 2 You can only combine
More informationUnit 1: Number System Fluency
Unit 1: Number System Fluency Choose the best answer. 1. Represent the greatest common factor of 36 and 8 using the distributive property. 36 + 8 = A 4 x (9 + 2) C 8 x (5+2) B 2 x (18+4) D 11 x (3+1) 2.
More informationThis is Solving Linear Systems, chapter 3 from the book Advanced Algebra (index.html) (v. 1.0).
This is Solving Linear Systems, chapter 3 from the book Advanced 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 informationRe: January 27, 2015 Math 080: Final Exam Review Page 1 of 6
Re: January 7, 015 Math 080: Final Exam Review Page 1 of 6 Note: If you have difficulty with any of these problems, get help, then go back to the appropriate sections and work more problems! 1. Solve for
More information2.1 Simplifying Algebraic Expressions
.1 Simplifying Algebraic Expressions A term is a number or the product of a number and variables raised to powers. The numerical coefficient of a term is the numerical factor. The numerical coefficient
More information15) x3/2 = ) (5x + 3)1/3 = 3. 17) (x2 + 14x + 49) 3/4-20 = 7. 18) x4-7x = 0. 19) x2/5 - x1/5-12 = 0. 21) e2x + ex - 6 = 0
Instructor: Medina Solve the equation. 1) x 9 = x 4 + 7 9 Name 15) x3/2 = 125 2) x + 7 4 = 2 - x - 1 6 16) (5x + 3)1/3 = 3 17) (x2 + 14x + 49) 3/4-20 = 7 3) 7 x = 1 2x + 52 4) 30 x - 4 + 5 = 15 x - 4 18)
More informationWhat value of x satisfies the equation = 1? 2
Math 8 EOG Review Problems Name # If a question is marked, you need to solve the problem without using a calculator. If a problem is not marked, you may use a calculator to solve to the problem. If a question
More information4. Smaller cylinder: r = 3 in., h = 5 in. 6. Let 3x the measure of the first angle. Let x the measure of the second angle.
Chapter : Linear Equations and Inequalities in One Variable.6 Check Points. A, b A bh h h h The height of the sail is ft.. Use the formulas for the area and circumference of a circle. The radius is 0 ft.
More information7) 24% of the lawyers in a firm are female. If there are 150 lawyers altogether, how many lawyers are female?
Math 110 Sample Final Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Find the perimeter (or circumference) and area of the figure. 1) Give the exact
More informationa 17 9 d rt 700 (72)t P 2L 2W 19 2L 2(3) L 13 A LW 50 12W W 50 9a a a a 17 9a / 9 17 / 9 9a
Free Pre-Algebra Lesson 0 page Lesson 0 Equations with Fractions Here we practice solving equations that involve fractions. The Solution is A Fraction Even equations with all whole numbers can have a fraction
More informationName: 1. 2,506 bacteria bacteria bacteria bacteria. Answer: $ 5. Solve the equation
Name: Print Close During a lab experiment, bacteria are growing continuously at an exponential rate. The initial number of bacteria was 120, which increased to 420 after 5 days. If the bacteria continue
More informationCommon Core State Standards for Mathematics
A Correlation of To the Common Core State Standards for Mathematics Units Unit 1 - Understanding Equal Groups Unit 2 Graphs and Line Plots Unit 3 - Travel Stories and Collections Unit 4 - Perimeter, Area,
More informationMidterm Review Packet
Algebra 1 CHAPTER 1 Midterm Review Packet Name Date Match the following with the appropriate property. 1. x y y x A. Distributive Property. 6 u v 6u 1v B. Commutative Property of Multiplication. m n 5
More informationUNIT 2 SOLVING EQUATIONS
UNIT 2 SOLVING EQUATIONS NAME: GRADE: TEACHER: Ms. Schmidt _ Solving One and Two Step Equations The goal of solving equations is to. We do so by using. *Remember, whatever you to do one side of an equation.
More informationRate of Change and slope. Objective: To find rates of change from tables. To find slope.
Linear Functions Rate of Change and slope Objective: To find rates of change from tables. To find slope. Objectives I can find the rate of change using a table. I can find the slope of an equation using
More informationproportion, p. 163 cross product, p. 168 scale drawing, p. 170
REVIEW KEY VOCABULARY classzone.com Multi-Language Glossary Vocabulary practice inverse operations, p. 14 equivalent equations, p. 14 identity, p. 156 ratio, p. 162 proportion, p. 16 cross product, p.
More informationState whether the following statements are true or false: 27.
Cumulative MTE -9 Review This packet includes major developmental math concepts that students ma use to prepare for the VPT Math (Virginia Placement Test for Math or for students to use to review essential
More information1 1,059, ,210,144
Number and Operations in Base Ten 4: Fluently add and subtract multi-digit whole numbers using the standard algorithm. 1 I can fluently add and subtract multidigit numbers using the standard algorithm.
More informationMississippi College and Career Readiness Standards for Mathematics Scaffolding Document. Grade 6
Mississippi College and Career Readiness Standards for Mathematics Scaffolding Document Grade 6 Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems
More informationGCSE Mathematics Practice Tests: Set 1
GCSE Mathematics Practice Tests: Set 1 Paper 2H (Calculator) Time: 1 hour 30 minutes You should have: Ruler graduated in centimetres and millimetres, protractor, pair of compasses, pen, HB pencil, eraser,
More informationWriting Equations 2.1. Addition Subtraction Multiplication Division
Writing Equations 2.1 Addition Subtraction Multiplication Division Example 1 A number b divided by three is six less than c. Example 2 Fifteen more than z times 6 is 11 less than y times 2. Example 3 Twenty
More informationNote: In this section, the "undoing" or "reversing" of the squaring process will be introduced. What are the square roots of 16?
Section 8.1 Video Guide Introduction to Square Roots Objectives: 1. Evaluate Square Roots 2. Determine Whether a Square Root is Rational, Irrational, or Not a Real Number 3. Find Square Roots of Variable
More informationg( x) = 3x 4 Lesson 10 - Practice Problems Lesson 10 Rational Functions and Equations Practice Problems
Lesson 10 - Practice Problems Section 10.1: Characteristics of Rational Functions 1. Complete the table below. Function Domain a) f ( x) = 4x 6 2x b) f ( x) = 8x + 2 3x 9 c) s( t ) = 6t + 4 t d) p( t )
More informationChapter 1: Introduction to Physics
Answers to Even-Numbered Conceptual Questions. The quantity T + d does not make sense physically, because it adds together variables that have different physical dimensions. The quantity d/t does make
More informationArchdiocese of Washington Catholic Schools Academic Standards Mathematics
6 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compare and order positive and negative integers*, decimals, fractions, and mixed numbers. They find multiples*
More informationImportant: You must show your work on a separate sheet of paper. 1. There are 2 red balls and 5 green balls. Write the ratio of red to green balls.
Math Department Math Summer Packet: Incoming 8 th -Graders, 2018-2019 Student Name: Period: Math Teacher: Important: You must show your work on a separate sheet of paper. Remember: It is important to arrive
More informationMath 0200 Final Exam Review Questions
Math 000 Final Eam Review Questions 1. Simplif: 4 8i + 8 ( 7). Simplif: 11 ( 9) + 6(10 4) + 4. Simplif ( 5 + 7) ( ) 8 6 4. Simplif: (4 ) 9 i 5. Simplif: 4 7 6. Evaluate 4 + 5 when = and = Write each of
More informationStudy Guide for Exam 2
Math 152 A Intermediate Algebra Fall 2012 Study Guide for Exam 2 Exam 2 is scheduled for Thursday, September 20"^. You may use a 3" x 5" note card (both sides) and a scientific calculator. You are expected
More informationState whether the following statements are true or false: 30. 1
Cumulative MTE -9 Review This packet includes major developmental math concepts that students ma use to prepare for the VPT Math (Virginia Placement Test for Math or for students to use to review essential
More informationName. 1. Given the solution (3, y), what is the value of y if x + y = 6? 7. The graph of y = x 2 is shown below. A. 3 B. 4 C. 5 D.
Name 1. Given the solution (, y), what is the value of y if x + y = 6? 7. The graph of y = x is shown below. 5 D. 6. What are the solutions to the equation x - x = 0? x = - or x = - x = - or x = 1 x =
More informationLesson 1. Unit 6 Practice Problems. Problem 1. Solution
Unit 6 Practice Problems Lesson 1 Lesson 2 Lesson 3 Lesson 4 Lesson 5 Lesson 6 Lesson 7 Lesson 8 Lesson 9 Lesson 10 Lesson 11 Lesson 12 Lesson 13 Lesson 14 Lesson 15 Lesson 16 Lesson 17 Lesson 18 Lesson
More informationMATH 410 Notes Simplifying Algebraic Expressions
MATH 410 Notes 2015 1.9 - Simplifying Algebraic Expressions Commutative Property: a + b = b + a and a b = b a Associative Property: a + (b + c) = (a + b) + c and a (b c) = (a b) c Distributive Property:
More informationIntermediate Algebra Semester Summary Exercises. 1 Ah C. b = h
. Solve: 3x + 8 = 3 + 8x + 3x A. x = B. x = 4 C. x = 8 8 D. x =. Solve: w 3 w 5 6 8 A. w = 4 B. w = C. w = 4 D. w = 60 3. Solve: 3(x ) + 4 = 4(x + ) A. x = 7 B. x = 5 C. x = D. x = 4. The perimeter of
More information1. Write in symbols: (a) The quotient of -6 and the sum of 2 and -8. (b) Now Simplify the expression in part a. 2. Simplify. x 4, given x=-2 and y=4
Sample problems for common Final Exam Math 115 LASC Directions: To receive credit show enough work so that your method of solution is clear. Box answers. Show all work on this test form. No Work=No Credit.
More informationMATH 110: FINAL EXAM REVIEW
MATH 0: FINAL EXAM REVIEW Can you solve linear equations algebraically and check your answer on a graphing calculator? (.) () y y= y + = 7 + 8 ( ) ( ) ( ) ( ) y+ 7 7 y = 9 (d) ( ) ( ) 6 = + + Can you set
More informationThe Celsius temperature scale is based on the freezing point and the boiling point of water. 12 degrees Celsius below zero would be written as
Prealgebra, Chapter 2 - Integers, Introductory Algebra 2.1 Integers In the real world, numbers are used to represent real things, such as the height of a building, the cost of a car, the temperature of
More informationDue for this week. Slide 2. Copyright 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
MTH 09 Week 1 Due for this week Homework 1 (on MyMathLab via the Materials Link) The fifth night after class at 11:59pm. Read Chapter 6.1-6.4, Do the MyMathLab Self-Check for week 1. Learning team coordination/connections.
More informationLesson 4: PILI* Numerical Reasoning
Lesson 4: PILI* Numerical Reasoning SECTION BREAKDOWN, SAMPLE QUESTIONS, TIPS Disclaimer: 12minprep is not affiliated, nor belongs to PI, which are the owners of Predictive Index Learning Indicator (PILI)
More information