## Solving Systems Of Linear Equations By Substitution 52 Exercises

**Vocabulary and Core Concept Check**Question 1.Describe how to solve a system of linear equations by substitution.Answer:

Question 2.**NUMBER SENSE**When solving a system of linear equations by substitution, how do you decide which variable to solve for in Step 1?Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3â8, tell which equation you would choose to solve for one of the variables. Explain.**Question 3.

Question 17.**ERROR ANALYSIS**Describe and correct the error in solving for one of the variables in the linear system 8x + 2y = -12 and 5x â y = 4.Answer:

Question 18.**ERROR ANALYSIS**Describe and correct the error in solving for one of the variables in the linear system 4x + 2y = 6 and 3x + y = 9.Answer:

Question 19.**MODELING WITH MATHEMATICS**A farmer plants corn and wheat on a 180-acre farm. The farmer wants to plant three times as many acres of corn as wheat. Write a system of linear equations that represents this situation. How many acres of each crop should the farmer plant?Answer:

Question 20.**MODELING WITH MATHEMATICS**A company that offers tubing trips down a river rents tubes for a person to use and âcoolerâ tubes to carry food and water. A group spends $270 to rent a total of 15 tubes. Write a system of linear equations that represents this situation. How many of each type of tube does the group rent?Answer:

**In Exercises 21â24, write a system of linear equations that has the ordered pair as its solution.**Question 21.

## Solving Systems Of Linear Equations Study Skills: Analyzing Your Errors

**5.1 â 5.4 What Did You Learn?**

**Core Vocabulary**system of linear equations, p. 236solution of a system of linear equations, p. 236

**Core Concepts**Solving a System of Linear Equations by Graphing, p. 237

Section 5.2Solving a System of Linear Equations by Substitution, p. 242

Section 5.3Solving a System of Linear Equations by Elimination, p. 248

Section 5.4Solutions of Systems of Linear Equations, p. 254

**Mathematical Practices**

Question 1.Describe the given information in Exercise 33 on page 246 and your plan for finding the solution.Answer:

Question 2.Describe another real-life situation similar to Exercise 22 on page 251 and the mathematics that you can apply to solve the problem.Answer:

Question 3.What question can you ask your friend to help her understand the error in the statement she made in Exercise 32 on page 258?Answer:

**Study Skills: Analyzing Your Errors**

**Study Errors**What Happens: You do not study the right material or you do not learn it well enough to remember it on a test without resources such as notes.How to Avoid This Error: Take a practice test. Work with a study group. Discuss the topics on the test with your teacher. Do not try to learn a whole chapterâs worth of material in one night.

## Lesson 55 Solving Equations By Graphing

**Essential Question** How can you use a system of linear equations to solve an equation with variables on both sides?

Previously, you learned how to use algebra to solve equations with variables on both sides. Another way is to use a system of linear equations.

**EXPLORATION 1**

Solving an Equation by GraphingWork with a partner. Solve 2x â 1 = â \ x + 4 by graphing.a. Use the left side to write a linear equation. Then use the right side to write another linear equation.b. Graph the two linear equations from part . Find the x-value of the point of intersection. Check that the x-value is the solution of2x â 1 = â\x + 4.c. Explain why this âgraphical methodâ works.

**EXPLORATION 2**

Solving Equations Algebraically and GraphicallyWork with a partner. Solve each equation using two methods.Method 1 Use an algebraic method.Method 2 Use a graphical method.Is the solution the same using both methods?a. \x + 4 = â\x + 1b. \x + 4 = \x + 3c. â\ x â 1 = \x â 4d. \ x + \ = 3x â 3e. -x + 2.5 = 2x â 0.5f. â 3x + 1.5 = x + 1.5

**Communicate Your Answer**

Question 3.How can you use a system of linear equations to solve an equation with variables on both sides?Answer:

Question 4.Compare the algebraic method and the graphical method for solving a linear equation with variables on both sides. Describe the advantages and disadvantages of each method.Answer:

**Solve the equation by graphing. Check your solution.**Question 1.\x â 3 = 2xAnswer:

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## Solving Systems Of Linear Equations 5154 Quiz

**Use the graph to solve the system of linear equations. Check your solution.**Question 1.y = â \x + 2y = x â 2

Answer:

Question 13.You plant a spruce tree that grows 4 inches per year and a hemlock tree that grows 6 inches per year. The initial heights are shown.a. Write a system of linear equations that represents this situation.b. Solve the system by graphing. Interpret your solution.Answer:

Question 14.It takes you 3 hours to drive to a concert 135 miles away. You drive 55 miles per hour on highways and 40 miles per hour on the rest of the roads.a. How much time do you spend driving at each speed?b. How many miles do you drive on highways? the rest of the roads?Answer:

Question 15.In a football game, all of the home teamâs points are from 7-point touchdowns and 3-point field goals. The team scores six times. Write and solve a system of linear equations to find the numbers of touchdowns and field goals that the home team scores.Answer:

## Lesson 57 Systems Of Linear Inequalities

**Essential Question **How can you graph a system of linear inequalities?

**EXPLORATION 1**

Work with a partner. Match each linear inequality with its graph. Explain your reasoning.2x + y â¤ 4 Inequality 12x â y â¤ 0 Inequality 2

**EXPLORATION 2**

Graphing a System of Linear InequalitiesWork with a partner. Consider the linear inequalities given in Exploration 1.2x + y â¤ 4 Inequality 12x â y â¤ 0 Inequality 2a. Use two different colors to graph the inequalities in the same coordinate plane. What is the result?b. Describe each of the shaded regions of the graph. What does the unshaded region represent?

**Communicate Your Answer**

How can you graph a system of linear inequalities?Answer:

When graphing a system of linear inequalities, which region represents the solution of the system?Answer:

Do you think all systems of linear inequalities have a solution? Explain your reasoning.Answer:

Write a system of linear inequalities represented by the graph.Answer:

**Tell whether the ordered pair is a solution of the system of linear inequalities.**Question 1. y < 5 y > x â 4Answer:

y â¥ 3x + 1y > x â 1

**Graph the system of linear inequalities.**Question 3.

Name another solution of Example 6.Answer:

Question 9.**WHAT IF?**You want to spend at least 3 hours at the mall. How does this change the system? Is still a solution? Explain.Answer:

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## Big Ideas Math Book Algebra 1 Answer Key Chapter 5 Solving Systems Of Linear Equations

Here is the complete list of topic-wise Big Ideas Math Book Algebra 1 Chapter 5 Solving Systems of Linear Equations Solution Key which covers the questions from the BIM Textbooks based on the latest Common Core Curriculum. **Ch 5 Big Ideas Math Textbook Algebra 1 Answers** material given here offers Questions from Exercises, Chapter Tests, Review Tests, Quiz, Assessment Tests, Cumulative Assessments, etc. Practice thoroughly and gain more subject knowledge on the concepts of BIM Math Book Algebra 1 Chapter 5 Solutions.

## Solving Systems Of Linear Equations Chapter Review

**5.1 Solving Systems of Linear Equations by Graphing **

**Solve the system of linear equations by graphing.**Question 1.

x + y < 1 5x + y > 4Answer:

y â¥ â \x + 1-3x + y > -2Answer:

Question 12.You pay $45.50 for 10 gallons of gasoline and 2 quarts of oil at a gas station. Your friend pays $22.75 for 5 gallons of the same gasoline and 1 quart of the same oil.a. Is there enough information to determine the cost of 1 gallon of gasoline and 1 quart of oil? Explain.b. The receipt shown is for buying the same gasoline and same oil. Is there now enough information to determine the cost of 1 gallon of gasoline and 1 quart of oil? Explain.c. Determine the cost of 1 gallon of gasoline and 1 quart of oil.Answer:

Describe the advantages and disadvantages of solving a system of linear equations by graphing.Answer:

Question 14.You have at most $60 to spend on trophies and medals to give as prizes for a contest.a. Write and graph an inequality that represents the numbers of trophies and medals you can buy. Identify and interpret a solution of the inequality.b. You want to purchase at least 6 items. Write and graph a system that represents the situation. How many of each item can you buy?Answer:

Question 15.Compare the slopes and y-intercepts of the graphs of the equations in the linear system 8x + 4y = 12 and 3y = -6x â 15 to determine whether the system has one solution, no solution, or infinitely many solutions. Explain.Answer:

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## Solving Equations By Graphing 55 Exercises

**Vocabulary and Core Concept Check**Question 1.**REASONING**The graphs of the equations y = 3x â 20 and y = -2x + 10 intersect at the point . Without solving, find the solution of the equation 3x â 20 = -2x + 10.Answer:

Question 2.**WRITING**Explain how to rewrite the absolute value equation |2x â 4| = |-5x + 1| as two systems of linear equations.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3â6, use the graph to solve the equation. Check your solution.**Question 3.

Question 33.**MODELING WITH MATHEMATICS**You need to hire a catering company to serve meals to guests at a wedding reception. Company A charges $500 plus $20 per guest. Company B charges $800 plus $16 per guest. For how many guests are the total costs the same at both companies?Answer:

Question 34.**MODELING WITH MATHEMATICS**Your dog is 16 years old in dog years. Your cat is 28 years old in cat years. For every human year, your dog ages by 7 dog years and your cat ages by 4 cat years. In how many human years will both pets be the same age in their respective types of years?Answer:

Question 35.**MODELING WITH MATHEMATICS**You and a friend race across a field to a fence and back. Your friend has a 50-meter head start. The equations shown represent you and your friendâs distances d from the fence t seconds after the race begins. Find the time at which you catch up to your friend.You: d = |-5t + 100|Your friend: d = |-3\t + 50|Answer:

f = -2x + 1 g = fAnswer:

f = \x â 2 g = fAnswer:

## Chapter 5 Algebra Equations Functions And Graphs

Consider the following statements and click to reveal the answer.

1. Simplify the following expressions:

**Answer:**

2. Find the general term for each of these sequences:

**Answer:**

3. Solve the following pairs of simultaneous equations:

**Answer:**

4. Find the gradients and the y-intercepts of the graphs of the following equations:

**Answer:**

5. Find the general term for the following sequence:

**Answer:**

6. What is a conjecture?

**Answer:**

*A conjecture is a hypothesis: something that has been surmised or deduced.*

7. What different methods can be used to solve simultaneous linear equations?

**Answer:**

*The methods that can be used to solve simultaneous linear equations are: *

*trial and improvement**eliminating the unknown by multiplying one of the equations to ensure that there is an equal number of a particular unknown in each equation**expressing one unknown in terms of another in one of the equations, then substituting the new expression into the other equation, which is then rearranged to find the solution.*

8. What is a function?

**Answer:**

*A function is a rule that changes or maps one number onto another, often represented in primary schools as a function machine.*

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## Solving Systems Of Linear Equations Cumulative Assessment

Question 1.The graph of which equation is shown?Answer:

Question 2.A van rental company rents out 6-, 8-, 12-, and 16-passenger vans. The function C = 100 + 5x represents the cost C of renting an x-passenger van for a day. Choose the numbers that are in the range of the function.Answer:

Question 3.Fill in the system of linear inequalities with < , â¤, > , or â¥ so that the graph represents the system.Answer:

Question 4.Your friend claims to be able to fill in each box with a constant so that when you set each side of the equation equal to y and graph the resulting equations, the lines will intersect exactly once. Do you support your friendâs claim? Explain.Answer:

Question 5.Select the phrases you should use when describing the transformations from the graph of f to the graph of g.Answer:

Which two equations form a system of linear equations that has no solution?Answer:

Question 7.Fill in a value for a so that each statement is true for the equation ax â 8 = 4 â x.Answer:

Which ordered pair is a solution of the linear inequality whose graph is shown?Answer:

Which of the systems of linear equations are equivalent?Answer:

Question 10.The value of x is more than 9. Which of the inequalities correctly describe the triangle? The perimeter is represented by P, and the area is represented by A.Answer:

## Systems Of Linear Inequalities 57 Exercises

**Vocabulary and Core Concept Check**Question 1.**VOCABULARY**How can you verify that an ordered pair is a solution of a system of linear inequalities?Answer:

Question 2.**WHICH ONE DOESNâT BELONG?**Use the graph shown. Which of the ordered pairs does not belong with the other three? Explain your reasoning.Answer:

**Monitoring Progress and Modeling with Mathematics**

**In Exercises 3â6, tell whether the ordered pair is a solution of the system of linear inequalities.**Question 3.

Question 29.**MODELING WITH MATHEMATICS**You can spend at most $21 on fruit. Blueberries cost $4 per pound, and strawberries cost $3 per pound. You need at least 3 pounds of fruit to make muffins.a. Write and graph a system of linear inequalities that represents the situation.b. Identify and interpret a solution of the system.c. Use the graph to determine whether you can buy 4 pounds of blueberries and 1 pound of strawberries.Answer:

Question 30.**MODELING WITH MATHEMATICS**You earn $10 per hour working as a manager at a grocery store. You are required to work at the grocery store at least 8 hours per week. You also teach music lessons for $15 per hour. You need to earn at least $120 per week, but you do not want to work more than 20 hours per week.a. Write and graph a system of linear inequalities that represents the situation.b. Identify and interpret a solution of the system.c. Use the graph to determine whether you can work 8 hours at the grocery store and teach 1 hour of music lessons.Answer:

**Mathematical Practices**

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## Lesson 54 Solving Special Systems Of Linear Equations

**Essential Question** Can a system of linear equations have no solution or infinitely many solutions?

**EXPLORATION 1**

Using a Table to Solve a SystemWork with a partner. You invest $450 for equipment to make skateboards. The materials for each skateboard cost $20. You sell each skateboard for $20.a. Write the cost and revenue equations. Then copy and complete the table for your cost C and your revenue R.b. When will your company break even? What is wrong?

**EXPLORATION 2**

Writing and Analyzing a SystemWork with a partner. A necklace and matching bracelet have two types of beads. The necklace has 40 small beads and 6 large beads and weighs 10 grams. The bracelet has 20 small beads and 3 large beads and weighs 5 grams. The threads holding the beads have no significant weight.a. Write a system of linear equations that represents the situation. Let x be the weight of a small bead and let y be the weight of a large bead.b. Graph the system in the coordinate plane shown. What do you notice about the two lines?c. Can you find the weight of each type of bead? Explain your reasoning.

**Communicate Your Answer**

Question 3.Can a system of linear equations have no solution or infinitely many solutions? Give examples to support your answers.Answer:

Question 4.Does the system of linear equations represented by each graph have no solution, one solution, or infinitely many solutions? Explain.Answer:

**Solve the system of linear equations.**Question 1.

## Big Ideas Math Algebra 1 Answers Chapter 5 Solving Systems Of Linear Equations

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