Amanda is decorating her new home and wants to buy some house plants. She is interested in three types of plants costing $4, $8, and $20. If she has budgeted exactly $96 for the plants and wants to buy exactly 12 of them, what are her options? Let x be the number of $4 plants, y the number of $8 plants, and z the number of $20 plants.

Question

Amanda is decorating her new home and wants to buy some house plants. She is interested in three types of plants costing $4, $8, and $20. If she has budgeted exactly $96 for the plants and wants to buy exactly 12 of them, what are her options? Let x be the number of $4 plants, y the number of $8 plants, and z the number of $20 plants.

Answer 1

here's one option:

z = 2
y = 4
x = 6

$20 * 2 = $40
$8 * 4 = $32
$4 * 6 = $24
$40 + $32 + $24 = $96
2 + 4 + 6 = 12
:)

Answer 2

To find Amanda's options, we need to solve the given system of equations:

1) x + y + z = 12 (as she wants to buy exactly 12 plants)
2) 4x + 8y + 20z = 96 (as she has budgeted exactly $96 for the plants)

We can use the method of substitution to solve this system.

First, rearrange equation 1 to find x in terms of y and z:
x = 12 - y - z

Substitute this value of x into equation 2:
4(12 - y - z) + 8y + 20z = 96

Simplify the equation:
48 - 4y - 4z + 8y + 20z = 96
-4y + 8y - 4z + 20z = 96 - 48
4y + 16z = 48

Divide both sides of the equation by 4:
y + 4z = 12

Now we have a new equation:
y + 4z = 12

We can rearrange this equation to find y in terms of z:
y = 12 - 4z

Now we have two equations:
1) x = 12 - y - z
2) y = 12 - 4z

We can substitute this value of y into equation 1 to find x in terms of z:
x = 12 - (12 - 4z) - z
x = 12 - 12 + 4z - z
x = 3z

Now we have three equations:
1) x = 3z
2) y = 12 - 4z
3) x + y + z = 12

We can substitute these equations into each other to find the values of x, y, and z.

Substitute value of x from equation 1 into equation 3:
3z + y + z = 12
4z + y = 12

Substitute the value of y from equation 2 into the above equation:
4z + (12 - 4z) = 12
4z - 4z + 12 = 12
12 = 12

This equation is true, indicating that there are infinitely many solutions for z.

To find the values of x, y, and z, we can assign a value to z and calculate the corresponding values of x and y.

Let's choose z = 0:
x = 3z = 3(0) = 0
y = 12 - 4z = 12 - 4(0) = 12

So, when z = 0, we have x = 0, y = 12.

Another option is to choose z = 3:
x = 3z = 3(3) = 9
y = 12 - 4z = 12 - 4(3) = 0

So, when z = 3, we have x = 9, y = 0.

Therefore, Amanda's options are:
Option 1: 0 plants costing $4, 12 plants costing $8, 0 plants costing $20.
Option 2: 9 plants costing $4, 0 plants costing $8, 3 plants costing $20.