Solving Linear Systems by Substitution Worksheet

**Page 1**

1.

A shopkeeper sold 32 softballs and basketballs last week for a total sum of $224. The price of a softball is $6 and that of a basketball is $14. How many softballs and basketballs did the shopkeeper sell?

a. | 5 softballs, 28 basketballs | ||

b. | 27 softballs, 5 basketballs | ||

c. | 28 softballs, 4 basketballs | ||

d. | 30 softballs, 30 basketballs |

[Equation 1.]

6

[Equation 2.]

[Rearrange equation 1.]

6(-

[Substitute the values.]

8

[Group the like terms.]

8

[Subtract 192 from the two sides of the equation.]

[Divide throughout by 8.]

[Substitute the values.]

The shopkeeper sold 28 softballs and 4 basketballs.

Correct answer : (3)

2.

Danielle purchased a total of 48 books and toys for the Boone play school. Each book costs $23 and each toy costs $6. How many books and toys did she buy for $730?

a. | 25 books and 23 toys | ||

b. | 23 books and 25 toys | ||

c. | 22 books and 26 toys | ||

d. | 26 books and 22 toys |

[Express as a linear equation.]

23

[Equation for the total cost of the books and toys.]

23

[From equation 1,

17

[Group the like terms.]

17

[Subtract 288 from the two sides of the equation.]

[Divide throughout by 17.]

[Substitute the values.]

Danielle bought 26 books and 22 toys.

Correct answer : (4)

3.

A circus company sold 201 tickets on a particular day. The entry fee is $22 for an adult and $7 for a child. The total amount collected was $2,802. How many adults and children went to the circus on that day?

a. | 108 adults and 93 children | ||

b. | 109 adults and 92 children | ||

c. | 92 adults and 109 children | ||

d. | 93 adults and 108 children |

Let

Total number of tickets = 201

So, Number of adults + Number of children = 201

[Equation 1.]

Total amount collected = $2802

So, (Cost of an adult ticket) × (Number of adults) + (Cost of a child ticket) × (Number of children) = $2802

22

[Substitute the values.]

22(-

[Substitute the values.]

-15

[Group the like terms.]

-15

[Subtract 4422 from the two sides of the equation.]

[Divide throughout by -15.]

[Substitute the values.]

93 adults and 108 children went to the circus on that day.

Correct answer : (4)

4.

A mechanical plant hires 1,199 labors on a daily wage scheme paying $9,691. Men are paid $9 and women are paid $7. Find the number of men and women hired.

a. | 649 men, 550 women | ||

b. | 550 men, 649 women | ||

c. | 551 men, 648 women | ||

d. | 648 men, 501 women |

Let number of women be

[Express as a linear equation.]

9

[Equation for the daily wages paid.]

[Rearrange equation 1.]

9

[Substitute the values.]

2

[Group the like terms.]

2

[Subtract 8393 from the two sides of the equation.]

[Divide throughout by 2.]

[Substitute the values.]

550 women and 649 men are hired.

Correct answer : (1)

5.

Find the value of $m$ such that the lines $y$ = 5$x$ - 1, $x$ = 4 and $y$ = $m$$x$ + 4 are concurrent.

a. | $\frac{15}{4}$ | ||

b. | $\frac{1}{19}$ | ||

c. | $\frac{19}{4}$ | ||

d. | $\frac{4}{15}$ |

[Given.]

The given equations are concurrent . So, they pass through the same point.

[Given.]

From equation (2),

Consider equation (1),

[Substitute

Consider equation (3),

[Substitute

[Multiply.]

[Subtract 4 from both sides.]

[Divide both sides by 4.]

Therefore, the value of

Correct answer : (1)

6.

Kelsey purchased a total of 31 books and toys for the Lake City play school. Each book costs $35 and each toy costs $9. How many books and toys did she buy for $747?

a. | 18 books and 13 toys | ||

b. | 28 books and 59 toys | ||

c. | 31 books and 1 toy | ||

d. | 21 books and 83 toys |

[Linear equation for the total books and toys.]

35

[Equation for the total cost of the books and toys.]

35

[From equation 1,

26

[Combine like terms.]

26

[Subtract 279 from each side.]

[Divide each side by 26.]

[Substitute

Kelsey bought 18 books and 13 toys.

Correct answer : (1)

7.

Solve the system by substitution.

4$x$ + $y$ = 5

- 16$x$ - 4$y$ = - 20

a. | (20.3, -76.2) | ||

b. | no solution | ||

c. | infinite solutions | ||

d. | (1, 1) |

[First equation.]

[Solve for

- 16

[Second equation.]

- 16

[Substitute the values.]

- 16

0 = 0

[Always true]

So, the number of solutions is infinite.

The solution is {(

Correct answer : (3)

8.

Solve the system by substitution:

10$x$ + 10$y$ = 100

12$x$ = - 12$y$ + 108

a. | (100, 0) | ||

b. | No solution | ||

c. | (100, 10) | ||

d. | {($x$, $y$): 10$x$ + 10$y$ = 100} |

[Second equation.]

[Divide throughout by 12 .]

10

[First equation.]

10( -

[Substitute the values.]

- 10

90 = 100

[This is a contradiction.]

So, there is no solution and the system is inconsistent.

Correct answer : (2)

9.

Find the value of $a$ for the given dependent system.

$y$ = 5$x$ + $a$

30$x$ - 6$y$ = 54

a. | - 9 | ||

b. | 55 | ||

c. | 54 | ||

d. | 6 |

[First equation.]

30

[Second equation.]

30

[Substitute the values.]

- 6

[Divide throughout by - 6.]

Correct answer : (1)

10.

Find the value of $a$ for the given dependent system.

6$y$ = 7$x$

30$y$ - 41$a$ - 35$x$ = 0

a. | 13 | ||

b. | 8 | ||

c. | 9 |

[First equation.]

[Simplify.]

30

[Second equation.]

30(

[Substitute the values.]

35

41

Correct answer : (2)