Question

systems of equations word problems directions: solve each system of equation word problems. then find your answer in the box on the back and color the numbered spaces using the indicated color. 1. taylor swift is selling digital copies and hard copies of her newest album. if she sells 13 hard copies and 7 digital copies, the total is $107. if taylor sells 7 hard copies and 7 digital copies, the total is $77. find the cost of taylor swift’s digital album. 2. the sum of two numbers is 24. the difference of the same two numbers is 2. find the value of the larger number. 3. arianna has a total of 34 nickels and dimes. together the coins total $2.05. how many nickels does arianna have? 4. two groups of friends go to the carnival. in one group 5 people buy fried oreos and 7 people buy funnel cakes for a total of $38.25. in the other group, 11 people buy funnel cakes and 4 people buy fried oreos for a total of $49.50. how much is a funnel cake? 5. the sum of two numbers is 43. the larger number is 7 more than twice the smaller. find the smaller number. 6. ms. daisy and ms. rose decide to spruce up their gardens and go shopping at the same plant store. ms. daisy spends $177 on 3 rose bushes and 13 tulips. ms. rose spends $247 for 13 rose bushes and 13 tulips. find the cost of one rose bush. 7. opening night of the musical at finn’s school sells 52 child tickets and 13 adult tickets for a total of $767. on the second night, the musical sold 40 child tickets and 16 adult tickets for a total of $668. find the cost of a child ticket. 8. justin timberlake needs new microphones and speakers for his next concert. for a total of $291, he can buy 6 speakers and 3 microphones. if he buys 9 speakers and 11 microphones then the total is $586. find the cost of one microphone.

Explanation:

Problem 1

Step1: Define variables

Let $x$ = cost of hard copy, $y$ = cost of digital copy.

Step2: Set up equations

$$13x + 7y = 107$$
$$7x + 7y = 77$$

Step3: Subtract equations

Subtract the second equation from the first:
$$(13x+7y)-(7x+7y)=107-77$$
$$6x=30$$

Step4: Solve for $x$

$$x=\frac{30}{6}=5$$

Step5: Solve for $y$

Substitute $x=5$ into $7x+7y=77$:
$$7(5)+7y=77$$
$$35+7y=77$$
$$7y=42$$
$$y=6$$

Problem 2

Step1: Define variables

Let $x$ = larger number, $y$ = smaller number.

Step2: Set up equations

$$x+y=24$$
$$x-y=2$$

Step3: Add equations

Add the two equations:
$$(x+y)+(x-y)=24+2$$
$$2x=26$$

Step4: Solve for $x$

$$x=\frac{26}{2}=13$$

Problem 3

Step1: Define variables

Let $x$ = number of nickels, $y$ = number of dimes.

Step2: Set up equations

$$x+y=34$$
$$0.05x+0.10y=2.05$$

Step3: Isolate $y$ from first equation

$$y=34-x$$

Step4: Substitute into second equation

$$0.05x+0.10(34-x)=2.05$$
$$0.05x+3.4-0.10x=2.05$$
$$-0.05x=2.05-3.4$$
$$-0.05x=-1.35$$

Step5: Solve for $x$

$$x=\frac{-1.35}{-0.05}=27$$

Problem 4

Step1: Define variables

Let $x$ = cost of fried Oreos, $y$ = cost of funnel cake.

Step2: Set up equations

$$5x+7y=38.25$$
$$4x+11y=49.50$$

Step3: Eliminate $x$

Multiply first equation by 4, second by 5:
$$20x+28y=153$$
$$20x+55y=247.5$$
Subtract first new equation from second:
$$(20x+55y)-(20x+28y)=247.5-153$$
$$27y=94.5$$

Step4: Solve for $y$

$$y=\frac{94.5}{27}=3.5$$

Problem 5

Step1: Define variables

Let $x$ = smaller number, $y$ = larger number.

Step2: Set up equations

$$x+y=43$$
$$y=2x+7$$

Step3: Substitute $y$ into first equation

$$x+(2x+7)=43$$
$$3x+7=43$$

Step4: Solve for $x$

$$3x=43-7=36$$
$$x=\frac{36}{3}=12$$

Problem 6

Step1: Define variables

Let $x$ = cost of rose bush, $y$ = cost of tulip.

Step2: Set up equations

$$3x+13y=177$$
$$13x+13y=247$$

Step3: Subtract equations

Subtract first equation from the second:
$$(13x+13y)-(3x+13y)=247-177$$
$$10x=70$$

Step4: Solve for $x$

$$x=\frac{70}{10}=7$$

Problem 7

Step1: Define variables

Let $x$ = cost of child ticket, $y$ = cost of adult ticket.

Step2: Set up equations

$$52x+13y=767$$
$$40x+16y=668$$

Step3: Simplify equations

Divide first equation by 13: $4x+y=59$ → $y=59-4x$
Divide second equation by 4: $10x+4y=167$

Step4: Substitute $y$

$$10x+4(59-4x)=167$$
$$10x+236-16x=167$$
$$-6x=167-236=-69$$

Step5: Solve for $x$

$$x=\frac{-69}{-6}=11.5$$

Problem 8

Step1: Define variables

Let $x$ = cost of speaker, $y$ = cost of microphone.

Step2: Set up equations

$$6x+3y=291$$
$$9x+11y=586$$

Step3: Simplify first equation

Divide by 3: $2x+y=97$ → $y=97-2x$

Step4: Substitute into second equation

$$9x+11(97-2x)=586$$
$$9x+1067-22x=586$$
$$-13x=586-1067=-481$$

Step5: Solve for $y$

$$x=\frac{-481}{-13}=37$$
Substitute $x=37$ into $y=97-2x$:
$$y=97-2(37)=97-74=23$$

Answer:

1. The cost of the digital album is $\$6$
2. The larger number is $13$
3. Arianna has 27 nickels
4. A funnel cake costs $\$3.50$
5. The smaller number is $12$
6. One rose bush costs $\$7$
7. A child ticket costs $\$11.50$
8. One microphone costs $\$23$