chapter 4 solving systems of linear equations and inequalities · pdf filechapter 4 solving...

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Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

4.1 Introduction to Systems of Linear Equations: Solving by Graphing Objectives A Decide whether an ordered pair is a solution of a system of linear equations in two

variables. B Determine the number of solutions of a system of linear equations. C Solve a system of linear equations by graphing. D Solve applied problems involving systems of linear equations. MATHEMATICALLY SPEAKING In exercises 1–4 fill in the blank with the most appropriate term or phrase from the given list.

are parallel graph set of equations

coincide system of equations solution 1. If the system has infinitely many solutions, the lines ___________________. 2. A(n) ___________________ of a system of two equations in two variables is an ordered

pair of numbers that makes both equations in the system true. 3. If the system has no solutions, the lines ___________________. 4. A(n) ___________________ is a group of two or more equations solved simultaneously. EXAMPLES AND PRACTICE Review this example for Objective A: Decide whether an ordered pair is a solution of a system of linear equations in two variables. 1. Is (2, 1) a solution of the system? 2

3 4

x y

x y

+ =− =

Substitute the x-coordinate 2 for x and the y-coordinate 1 for y in the equations and check if both equations are true.

?

?

2 2 1 2 3 2 False

3 4 3(2) 1 4 5 4 False

x y

x y

+ = → + = → =

− = → − = → =

Practice: 1. Is (2, 0) a solution of the system? 2 4

2

x y

x y

+ =− =



The ordered pair (2, 1) is not a solution of the system because it is does not satisfy both equations.

Review this example for Objective B: Determine the number of solutions of a system of linear equations. 2. Determine the number of solutions for the

system graphed.

The two lines in this graph intersect at a single point. So the system has one solution.

Practice: 2. Determine the number of solutions for

the system graphed.

Review this example for Objective C: Solve a system of linear equations by graphing. 3. Solve the following system by graphing: 1

5

y x

x y

= −+ = −

Graph each linear equation by using the x- and y-intercept method and then sketch the line that passes through these points. 1y x= − 5x y+ = −

x y x y 0 1− 0 5− 1 0 5− 0 4 3 1 6− Plot the points, and graph both equations.

Practice: 3. Solve the following system by graphing: 2

2

y x

x y

= −+ =



The lines appear to intersect at the point ( 2, 3).− − Check: Confirm that ( 2, 3)− − is the solution by substituting these values into the original equations:

?

?

1 3 2 1

3 3 True

5 2 ( 3) 5

5 5 True

y x

x y

= − → − =− −− = −

+ = − → − + − =−− = −

So ( 2, 3)− − is the solution of the system.

Review this example for Objective D: Solve applied problems involving systems of linear equations. 4. A plane flying with a tail wind, flew at a

speed of 550 mph, relative to the ground. When flying against the tail wind, it flew at a speed of 500 mph. Express these relationships as equations. Find the speed of the plane in calm air and the speed of the wind.

Let x represent the speed of the plane, and let y represent the speed of the wind. The given information can be expressed as:

550

500

x y

x y

+ =− =

Choose an appropriate scale and graph both equations.

Practice: 4. Bill the plumber charges $80 for a house

call and then $45 per hour for labor. Sue the plumber charges $65 for a house call and then $50 per hour for labor. Write a cost equation for each plumber, where y is the total cost of plumbing repairs and x is the number of hours of labor. For how many hours of labor would Bill and Sue charge the same amount?



The lines appear to intersect at (525, 25). Verify that this is the solution by substituting the values into both of the equations. The solution is the speed of the plane in calm air is 525 mph, and the speed of the wind is 25 mph.

ADDITIONAL EXERCISES Objective A Decide whether an ordered pair is a solution of a system of linear

equations in two variables. Indicate whether each ordered pair is or is not a solution to the given system. 1. 5 3 3

4 2 10

x y

x y

− =− =

for (10, 3) 2. 2 2 8

6 3

x y

x y

+ =− =

for (1, 3)



Objective B Determine the number of solutions of a system of linear equations. For each system graphed, determine the number of solutions. 3. 4.

Objective C Solve a system of linear equations by graphing. Solve by graphing. 5. 6

0

x y

x y

− =+ =

6. 2 2

3

x y

x y

+ = −+ = −

7. 2

2

x y

y x

+ == − −

8. 2 3

2 3

x y

y x

− = −= −



9. 3

1

y x

y x

= += − −

10. 3y x

y x

= −=

11. 4 2

1 1

4 2

x y

y x

+ =

= − +

12. 2

2 1

x y

x y

+ = −− = −



13. 2 1

2

y x

y x

= += +

14. 1

2 5

x y

y x

+ == −

15. 2 3

2 2

y x

y x

= += −

16. 2

2 2 4

y x

x y

= +− = −



Objective D Solve applied problems involving systems of linear equations. Solve. 17. A movie fan rented 5 films at a local video store. The daily rental charge was $3 on

some films and $5 on others. If the total rental charge was $17, how many $5 films were rented?




4.2 Solving Systems of Linear Equations by Substitution Objectives A Solve a system of linear equations by substitution. B Solve applied problems involving systems of linear equations. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear equations by substitution. 1. Solve by substitution: 3 2 6

1

x y

x y

+ = −+ = −

First solve for x or y in either of the equations. Let’s solve for y in the second equation.

1

1

x y

y x

+ = −= − −

Next, substitute the expression 1x− − for y in the first equation and solve for x.

3 2 6

3 2( 1) 6

3 2 2 6

2 6

4

x y

x x

x x

x

x

+ = −+ − − = −

− − = −− = −

= −

Solve for y by substituting 4− for x in the original second equation.

1

4 1

3

x y

y

y

+ = −− + = −

=

So the solution is ( 4, 3).− Check this in the original system.

Practice: 1. Solve by substitution: 2 3

3 6 4

x y

x y

+ =+ = −



Review this example for Objective B: Solve applied problems involving systems of linear equations. 2. During a sale, a store sells red-dot items

at a 35% discount and yellow-dot items at 25% discount. A shopper bought red- and yellow-dot items with a combined regular price of $57. If the total discount was $16.71, how much did the shopper spend on each kind of item?

Let r represent the regular price of the red-dot items purchased and y represent the regular price of the yellow-dot items purchased. The equation representing the combined regular price of the items purchased is 57r y+ = The equation representing the total discount is 0.35 0.25 16.71r y+ = The system of equations is

57

0.35 0.25 16.71

r y

r y

+ =+ =

Solve the first equation for y:

57

57

r y

y r

+ == −

Now substitute 57 r− for y in the second equation:

0.35 0.25 16.71

0.35 0.25(57 ) 16.71

0.35 14.25 0.25 16.71

0.1 2.46

24.6

r y

r r

r r

r

r

+ =+ − =

+ − ===

Solve for y by substitution 24.6 for r in the first original equation.

57

24.6 57

32.4

r y

y

y

+ =+ =

=

The solution of the system is 24.6r = and 32.4.y = In the context of this

Practice: 2. On a particular airline route, a full-price

coach ticket costs $350 and a discounted coach ticket costs $250. On one of these flights, there were 158 passengers in coach, which resulted in a total ticket income of $49,600. How many full-price tickets were sold?



problem, the shopper spent 0.65(24.6) $15.99= on red-dot items and 0.75(32.4) $24.30= on yellow-dot

items. ADDITIONAL EXERCISES Objective A Solve a system of linear equations by substitution. Solve by substitution and check. 1. 6 32

3

x y

y x

− == − +

2. 3 10

2

y x

y x

= − +=



3. 3

2 7

x y

x y

− == +

4. 2 1

1

y x

y

+ == −

5. 3 7 8

4 1

x y

x y

− = −− = −

6. 2 1

3 6 3

x y

x y

− =− =



7. 3 2

3 9 2

x y

x y

− =− =

8. 5 4 1

2 1

x y

x y

− = −− =

9. 3 2 3

11

x y

x y

− =+ =

10. 4 2 6

2 3

x y

y x

+ == − +



11. 7 8 0

2 0

x y

x y

+ =− =

12. 4 6

5 6

x y

x y

− =+ = −

Objective B Solve applied problems involving systems of linear equations. Solve. 13. A bottle of fruit juice contains 10% water. How much water must be added to this

bottle to produce 7 L of fruit juice that is 55% water?



14. A student took out two loans totaling $8000. She borrowed the maximum amount she could at 5% and the remainder at 6% interest per year. At the end of the first year, she owed $430 in interest. How much was loaned at each rate?

15. A $30,000 investment was split so that part was invested at 8% annual rate of interest

and the rest at 10%. If the total annual earnings were $2620, how much money was invested at each rate?




4.3 Solving Systems of Linear Equations by Elimination Objectives A Solve a system of linear equations by elimination. B Solve applied problems involving systems of linear equations. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear equations by elimination. 1. Solve the following system by the

elimination method. 3

7

x y

x y

+ = −− + =

Since the coefficients of the x-terms in the two equations are opposites, the x-terms are eliminated if we add the equations. 3

7

0 2 4

2 4

2

x y

x y

y

y

y

+ = −− + =

+ ===

Substitute 2 for y in either of the original equations. Substituting in the first equations we get:

3

2 3

5

x y

x

x

+ = −+ = −

= −

So 5x = − and 2.y = That is, the solution is ( 5, 2).− Check the solution in both original equations.

Practice: 1. Solve the following system by the

elimination method. 2

2 1

x y

x y

+ =− =



Review this example for Objective B: Solve applied problems involving systems of linear equations. 2. To enter a zoo, adult visitors must pay

$8, whereas children and seniors pay only half price. On one day the zoo collected a total of $1580. If the zoo had 246 visitors that day, how many half-price admissions and how many full-price admissions did the zoo collect?

Let f represent the number of full-priced admissions, and let h represent the number of half-priced admissions. We must solve the following system:

8 4 1580

246

f h

f h

+ =+ =

Eliminate the h-terms by multiplying the second equation by 4− and adding the equations. 8 4 1580

246 4 4 984

f h

f h f h

+ =+ = → − − = −

Add the equations:

8 4 1580

4 4 984

4 596

149

f h

f h

f

f

+ =− − = −

==

Substitute 149 for f in the original second equation and solve for h.

246

149 246

97

f h

h

h

+ =+ =

=

So on that particular day the zoo collected 149 full-priced admissions and 97 half-priced admissions.

Practice: 2. A crew team rows in a river with a

current. When the team rows with the current, the boat travels 16 miles in 2 hours. Against the current, the team rows 8 miles in the same amount of time. At what speed does the team row in still water?



ADDITIONAL EXERCISES Objective A Solve a system of linear equations by elimination. Solve. 1. 4

2 8

m n

m n

+ =− =

2. 2 3 4

4 6 3

x y

x y

+ =+ =

3. 2 1

2 3 3

x y

x y

+ =+ =

4. 4 3

3

x y

x y

− + =− =

5. 2 3 5

4 6 10

x y

x y

− =− =

6. 4 2 2

5 5

x y

x y

− = −− =



7. 8 3 2

5 2 1

x y

x y

+ =+ =

8. 2 2

2 5 2

x y

x y

− + =− = −

9. 4 5 6

4 5 1

x y

x y

− =− = −

10. 3 1

2 2 2

x y

x y

− =− + =



11. 5 4 5

6 3 3

x y

x y

+ = −+ =

12. 4 2

2 5 7

x y

x y

− =− =

13. 7 4 3

6 4 6

x y

x y

+ =+ =

14. 8 7 2

9 7 4

x y

x y

+ =+ =



Objective B Solve applied problems involving systems of linear equations. Solve. 15. The annual salaries of a congressman and a senator total $288,900. If the senator makes

$41,300 more than the congressman, find each of their salaries. 16. A novelty shop sells some embroidered scarves for $10 each and others for $14 each. A

customer pays $92 for 8 scarves. How many scarves at each price did she buy?




4.4 Solving Systems of Linear Inequalities Objectives A Solve a system of linear inequalities by graphing. B Solve applied problems involving systems of linear inequalities. MATHEMATICALLY SPEAKING In exercises 1–2 fill in the blank with the most appropriate term or phrase from the given list.

shaded regions triples boundary line pairs simultaneous both coordinate plane a system of

1. A solution to a system of two linear inequalities is a point that lies in both

___________________ of the graph. 2. The graph of the inequality 2 3 1x y+ ≤− includes the ___________________. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear inequalities by graphing. 1. Graph the solutions of the system:

3

12

31

12

y x

y x

y x

≤

<− +

> −

Begin by graphing each inequality on the same coordinate plane. Graph each boundary line, then for each inequality, shade the half-plane that contains its solutions. The solutions of the system are all the points that lie in the intersection of the shaded regions. Note that points on the line 3y x= are solutions but points

on the lines 1

23

y x=− + and

Practice: 1. Graph the solutions of the system:

2 4

11

23 2

y x

y x

y x

≥− +

<− −

≤ +



11

2y x= − are not.

Review this example for Objective B: Solve applied problems involving systems of linear inequalities. 2. The yearbook staff is selling pages of

space to students for the upcoming yearbook. A full page costs $125 and a half page costs $75. The yearbook must raise at least $2000 in revenue from these pages. More students bought full pages than half pages.

a. Express this information as a system of inequalities.

b. Graph the system. c. Give an example of the number of full

pages and the number of half pages the yearbook staff may have sold under the given conditions.

a. Let x represent the number of full pages sold, and y represent the number of half pages sold. Then the system to be solved is

125 75 2000x y

x y

+ ≥>

which can be rewritten as

5 80

3 3y x

y x

≥− +

<

Practice: 2. Carlo and Anita make mailboxes and toys

in a craft shop. Each mailbox, x, requires 1 hr of work from Carlo and 1 hr from Anita. Each toy, y, requires 1 hr of work from Carlo and 3 hr from Anita. Carlo can work no more than 7 hr per week and Anita can work no more than 15 hr per week.

a. Express this information as a system of inequalities.

b. Graph the system.

c. What does the solution region

represent?



b. Solve by graphing.

The solutions of the system are points that lie in the intersection of the two shaded regions, including part of the

boundary line 5 80

.3 3

y x=− +

c. The integer solutions in the shaded region represent all possible numbers of full pages and half pages the yearbook staff may have sold under the given conditions. One possible combination is 20 full pages and 10 half pages.

ADDITIONAL EXERCISES Objective A Solve a system of linear inequalities by graphing. Solve by graphing.

1. 8 3

4 5

y x

y x

< −<− +

2. 4 3

2 5

y x

y x

> −<− +



3. 3 9

3 9

x y

x y

− ≥+ <

4. 2 4

2 4

x y

x y

− ≥+ ≤

5. 5 2 10

5 3 15

x y

x y

+ <− <

6. 3 3 9

2 2 4

x y

x y

+ <− <

7. 3 1

3 4

y x

x y

< +− ≤

8. 2 1

2 6

y x

x y

< +− ≤



9. 3

4

y

x

>−≤

10. 2

43

3 1

y x

y x

≤− −

> +

11. 1.5 3

0.5 2

y x

y x

< −>− +

12.

4 5 10

2

5

x y

y

x

+ ≤≥−≥−

13.

2 3

2 3

2

y x

y x

x

≤ −≥− −≤

14.

3 3

3 3

2 0

y x

y x

x

≤ +>− +

− <



15.

2 4

2 1

2 0

y x

y x

x

≤ +>− −

− <

Objective B Solve applied problems involving systems of linear inequalities. Solve. 16. A college student works in both the school cafeteria and library. She works no

more than 10 hours per week in the cafeteria and no more than 17 hours per week in the library. She must work at least 20 hours each week. a. Express this information as a system of inequalities. b. Graph the system.

c. How many hours can she work in the library if she works 8 hours in the

cafeteria in one week?

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