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Solving Systems Using Substitution 589
Lesson 10-
When equations for lines in a system are in y = mx + b form, a method of solving called substitution can be very efficient. Example 1 illustrates this method.
Example 1
Solve the system
y = 7 x + 25 y = −5x - 11
using substitution.
Solution Because 7x + 25 and −5x - 11 both equal y, they must equal each other. Substitute one of them for y in the other equation. 7x + 25 = –5x - 11 Substitution 12x + 25 = –11 Add 5x to both sides. 12x = –36 Subtract 25 from both sides. x = –3 Divide both sides by 12. Now you know x = −3. However, you must still solve for y. You can substitute −3 for x into either of the original equations. We choose the first equation. y = 7x + 25 y = 7(–3) + 25 y = –21 + 25 y = 4 The solution is x = –3 and y = 4, or just (–3, 4).
QY
Check A graph shows that the lines with equations y = 7 x + 25 and y = −5x - 11 intersect at (−3, 4).
Suppose two quantities are increasing or decreasing at different constant rates. Then each quantity can be described by an equation of the form y = mx + b. To find out when the quantities are equal, you can solve a system using substitution. Example 2 illustrates this idea.
BIG IDEA Substituting an expression that equals a single variable is an effective first step for solving some systems.
QY Check that (−3, 4) is a solution to y = −5x - 11.
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y
x
y = 5 x - 11
y =^7 x +^25
Lesson
Solving Systems
10-2 Using Substitution
Find the greatest common factor. a. 15; 200 b. 1,500; 20, c. 14; 26; 53 d. 1,400; 2,600; 5,
Mental Math
590 Linear Systems
Example 2
The Rapid Taxi Company charges $2.15 for a taxi ride plus 20¢ for each __^1 10 mile traveled. A competitor, Carl’s Cabs, charges $1.50 for a taxi ride plus 25¢ for each __ 101 mile traveled. For what distance do the rides cost the same? Solution Let d = the distance of a cab ride in tenths of a mile. Let C = the cost of a cab ride of distance d. Rapid Taxi: C = 2.15 + 0.20d Carl’s Cabs: C = 1.50 + 0.25d The rides cost the same when the values of C and d for Rapid Taxi equal the values for Carl’s Cabs, so we need to solve the system formed by these two equations. Substitute 2.15 + 0.20d for C in the second equation. 2.15 + 0.20d = 1.50 + 0.25d Now solve. 0.65 = 0.05d Add −1.50 and −0.20d to both sides. d = 13 Divide both sides by 0.05. The two companies charge the same amount for a ride that is 13 tenths of a mile long, or 1.3 miles long. Check Check to see if the cost will be the same for a ride of 13 tenths of a mile. The cost for Rapid Taxi is 2.15 + 0.20 · 13 = 2.15 + 2.60 = 4.75. The cost for Carl’s Cabs is 1.50 + 0.25 · 13 = 1.50 + 3.25 = 4.75. The cost is $4.75 from each company, so the answer checks.
In Example 2, Carl’s Cabs is cheaper at first, but as the number of miles increases, the prices become closer. Eventually the price for Carl’s Cabs catches up with Rapid Taxi’s price, and then Carl’s is more expensive than Rapid. The next example also involves “catching up.”
Example 3
Bart was so confident that he could run faster than his little sister that he bragged, “I can beat you in a 50-meter race. I’m so sure that I’ll give you a 10-meter head start!” Bart could run at a speed of 4 meters per second, while his sister could run 3 meters per second. Could Bart catch up to his sister before the end of the race? Solution Let d be the distance that Bart and his sister have traveled after t seconds. Recall that distance = rate · time. For Bart, d = 4t.
Chapter 10
The average taxi fare in New York in 2006 was $9.65. Source: MSNBC
GUIDED
592 Linear Systems
- A tomato canning company has fixed monthly costs of $4,200. There are additional costs of $2.35 to produce each case of canned tomatoes. The company sells tomatoes to grocery stores for $5.85 per case. a. Write a system of equations to describe this situation. b. How many cases must the company sell to break even? c. Check your solution.
APPLYING THE MATHEMATICS
- A car leaves a gas station traveling at 60 mph. The driver has accidentally left his credit card at the gas station. Six minutes later, his friend leaves the station with the credit card, traveling at 65 mph to catch up to him. a. Write two equations to indicate the distance d that each car is from the gas station t hours after the first car leaves. b. Solve the system to determine when the second car will catch up to the first car. c. How far will they have traveled from the gas station when they meet?
- Cameron has $450 and saves $12 a week. Sean has only $290, but is saving $20 a week. a. After how many weeks will they each have the same amount of money? b. How much money will each person have then?
- In 2000, the metropolitan area of Dallas had about 5,200, people and was growing at about 120,000 people a year. In 2000, the metropolitan area of Boston had about 4,400,000 people and was growing at about 25,000 people a year. a. If these trends had been this way for quite some time, in what year did Dallas and Boston have the same population? b. What was this population?
- In July 2005, Philadelphia approved taxi fares with an initial charge of $2.30 and an additional charge of $0.30 for each __^17 mile. If P ( x ) is the cost for taking a taxi x miles, then P ( x ) = 2.30 + 0.30 · 7 x. In October 2005, Atlanta established new taxi fares with an initial charge of $2.50 and an additional charge of $0.25 for each __^18 mile. If A ( x ) is the cost of taking a taxi x miles in Atlanta, then A ( x ) = 2.50 + 0.25 · 8 x. Solve a system to approximate at what distance the fares for Philadelphia and Atlanta are the same.
Chapter 10
Approximately 124,900 acres of tomatoes were harvested in the United States in 2002. Source: U.S. Department of Agriculture
Solving Systems Using Substitution 593
- One plumbing company charges $55 for the first half hour of work and $25 for each additional half hour. Another company charges $ for the first half hour and then $30 for each additional half hour. For how many hours of work will the cost of each company be the same?
In 14 and 15, a system that involves a quadratic equation is given. Each system has two solutions. a. Solve the system by substitution. b. Check your answers.
y = 1 __ 9 x^2 y = 4 x
y = 2 x^2 + 5 x - 3 y = x^2 - 2 x + 5
REVIEW
- Consider the system
y = 20 x + 8 24 x - y = −
. Verify that (^) ( __^12 , 18 (^) ) is a
solution to the system, but that (1, 20) is not. (Lesson 10-1)
In 17 and 18, solve the system of equations by graphing. (Lesson 10-1)
- the system in Question 3
- the system in Question 4
- a. Simplify y ( y - 9) + 4 y + 1. b. Solve y ( y - 9) = 4 y + 1. (Lesson 9-5)
- Skill Sequence Solve each equation. (Lessons 9-1, 8-6)
a. n^2 = 16 b. √ n = 16 c. √ n^2 = 16
- What is the cost of x basketballs at $18 each and y footballs at $25 each? (Lessons 5-3, 1-2)
EXPLORATION
- Find the taxi rates where you live or in a nearby community. Graph the rates to show how they compare to those in Question 12.
Lesson 10-
QY ANSWER Does 4 = −5(−3) - 11? 4 = 15 - 11 4 = 4 Yes, it checks.
