The gardeners at Middleton Place Gardens want to plant a total of 45 white and pink hydrangeas in one flower bed. In another flower bed, they want 120 hydrangeas. In this bed, they want 2 times the number of white hydrangeas and 3 times the number of pink hydrangeas as in the first bed. Use a system of equations to find how many pink and how many white hydrangeas the gardeners should buy altogether.

Question

The gardeners at Middleton Place Gardens want to plant a total of 45 white and pink hydrangeas in one flower bed. In another flower bed, they want 120 hydrangeas. In this bed, they want 2 times the number of white hydrangeas and 3 times the number of pink hydrangeas as in the first bed. Use a system of equations to find how many pink and how many white hydrangeas the gardeners should buy altogether.

Answer 1

To approach this problem, let's define some variables to represent the number of hydrangeas we need. Let:

- x be the number of white hydrangeas in the first flower bed.
- y be the number of pink hydrangeas in the first flower bed.
- w be the number of white hydrangeas in the second flower bed.
- p be the number of pink hydrangeas in the second flower bed.

From the problem statement, we can determine the following equations:

In the first flower bed:
1) x + y = 45 (equation representing the total number of hydrangeas needed)

In the second flower bed:
2) w + p = 120 (equation representing the total number of hydrangeas needed)
3) w = 2x (equation representing the number of white hydrangeas being twice the number in the first bed)
4) p = 3y (equation representing the number of pink hydrangeas being thrice the number in the first bed)

Now, let's solve this system of equations:

From equation 3), we substitute w = 2x into equation 2):
2x + p = 120 (substituting w = 2x)

From equation 4), we substitute p = 3y into the above equation:
2x + 3y = 120 (substituting p = 3y)

We now have a system of two equations with two variables:
1) x + y = 45
2) 2x + 3y = 120

There are several methods to solve this system of equations. One option is to use substitution or elimination. Let's use the elimination method:

Multiply equation 1) by 2:
2(x + y) = 2(45)
2x + 2y = 90

Now, subtract equation 2) from the above equation:
(2x + 2y) - (2x + 3y) = 90 - 120
2x + 2y - 2x - 3y = -30
2y - 3y = -30
-y = -30

Multiply the entire equation by -1 to isolate y:
y = 30

Now, substitute the value of y into equation 1) to solve for x:
x + 30 = 45
x = 45 - 30
x = 15

So, the solution to the system of equations is x = 15 and y = 30.

Therefore, the gardeners should buy 15 white hydrangeas and 30 pink hydrangeas for a total of 45 hydrangeas in the first flower bed. In the second flower bed, they should buy 30 white hydrangeas and 90 pink hydrangeas for a total of 120 hydrangeas.