Question

1. micaela and shawna each improved their yards by planting daylilies and ivy. they bought their supplies from the same store. micaela spent $58 on 5 daylilies and 1 pot of ivy. shawna spent $90 on 5 daylilies and 5 pots of ivy. find the cost of one daylily and the cost of one pot of ivy.
2. gabriella and scott are selling pies for a school fundraiser. customers can buy cherry pies and pumpkin pies. gabriella sold 4 cherry pies and 1 pumpkin pie for a total of $64. scott sold 4 cherry pies and 13 pumpkin pies for a total of $256. find the cost each of one cherry pie and one pumpkin pie.
3. alberto and ryan each improved their yards by planting hostas and geraniums. they bought their supplies from the same store. alberto spent $13 on 2 hostas and 1 geranium. ryan spent $83 on 10 hostas and 7 geraniums. find the cost of one hosta and the cost of one geranium.
4. shawna and mark each improved their yards by planting rose bushes and ornamental grass. they bought their supplies from the same store. shawna spent $16 on 1 rose bush and 6 bunches of ornamental grass. mark spent $36 on 4 rose bushes and 10 bunches of ornamental grass. what is the cost of one rose bush and the cost of one bunch of ornamental grass?

Explanation:

Problem 42:

Step1: Define variables

Let \( x \) be the cost of one daylily (in dollars) and \( y \) be the cost of one pot of ivy (in dollars).

Step2: Set up equations

From Micaela's purchase: \( 5x + y = 58 \)
From Shawna's purchase: \( 5x + 5y = 90 \)

Step3: Subtract the first equation from the second

\( (5x + 5y) - (5x + y) = 90 - 58 \)
Simplify: \( 4y = 32 \)

Step4: Solve for \( y \)

Divide both sides by 4: \( y = \frac{32}{4} = 8 \)

Step5: Substitute \( y = 8 \) into the first equation

\( 5x + 8 = 58 \)
Subtract 8: \( 5x = 50 \)
Divide by 5: \( x = 10 \)

Step1: Define variables

Let \( c \) be the cost of one cherry pie (in dollars) and \( p \) be the cost of one pumpkin pie (in dollars).

Step2: Set up equations

From Gabriella's sales: \( 4c + p = 64 \)
From Scott's sales: \( 4c + 13p = 256 \)

Step3: Subtract the first equation from the second

\( (4c + 13p) - (4c + p) = 256 - 64 \)
Simplify: \( 12p = 192 \)

Step4: Solve for \( p \)

Divide both sides by 12: \( p = \frac{192}{12} = 16 \)

Step5: Substitute \( p = 16 \) into the first equation

\( 4c + 16 = 64 \)
Subtract 16: \( 4c = 48 \)
Divide by 4: \( c = 12 \)

Step1: Define variables

Let \( x \) be the cost of one hosta (in dollars) and \( y \) be the cost of one geranium (in dollars).

Step2: Set up equations

From Alberto's purchase: \( 2x + y = 13 \)
From Ryan's purchase: \( 10x + 7y = 83 \)

Step3: Solve the first equation for \( y \)

\( y = 13 - 2x \)

Step4: Substitute \( y = 13 - 2x \) into the second equation

\( 10x + 7(13 - 2x) = 83 \)
Expand: \( 10x + 91 - 14x = 83 \)
Simplify: \( -4x + 91 = 83 \)
Subtract 91: \( -4x = -8 \)
Divide by -4: \( x = 2 \)

Step5: Substitute \( x = 2 \) into \( y = 13 - 2x \)

\( y = 13 - 2(2) = 13 - 4 = 9 \)

Answer:

The cost of one daylily is \$10 and the cost of one pot of ivy is \$8.

Problem 43: