Answer :
Answer:
- Cost of one apple ( x ): $0.90
- Cost of one cantaloupe ( y ): $1.40
- Cost of one bunch of bananas ( z ): $2.70
Step-by-step explanation:
Let's denote:
- x as the cost of one apple,
- y as the cost of one cantaloupe, and
- z as the cost of one bunch of bananas.
We are given three sets of equations based on the purchases made over three weeks:
Two weeks ago:
[tex]\sf 10x + 8y + 13z = 55.30 [/tex]
Last week:
[tex]\sf 11x + 9y + 10z = 49.50 [/tex]
This week:
[tex]\sf 11x + 7y + 14z = 57.50 [/tex]
We can write this system of equations as follows:
[tex]\sf \begin{cases}10x + 8y + 13z = 55.30 \cdots (1)\\ 11x + 9y + 10z = 49.50 \cdots (2)\\ 11x + 7y + 14z = 57.50 \cdots (3) \\ \end{cases} [/tex]
To solve the system of equations, we can use the method of elimination. Let's start by manipulating the equations to eliminate one variable.
Subtract equation (1) from equation (2) to eliminate x :
[tex]\sf (11x + 9y + 10z) - (10x + 8y + 13z) = 49.50 - 55.30 [/tex]
Simplifying,
[tex]\sf x + y - 3z = -5.80 [/tex]
Subtract equation (1) from equation (3) to eliminate x :
[tex]\sf (11x + 7y + 14z) - (10x + 8y + 13z) = 57.50 - 55.30 [/tex]
Simplifying,
[tex]\sf x - y + z = 2.20 [/tex]
Now, we have a new system of equations:
[tex]\sf \begin{cases}x + y - 3z = -5.80 \\x - y + z = 2.20 \\\end{cases}[/tex]
Next, we can solve this simplified system by elimination or substitution.
Adding the two equations:
[tex]\sf (x + y - 3z) + (x - y + z) = -5.80 + 2.20 [/tex]
[tex]\sf 2x - 2z = -3.60 [/tex]
[tex]\sf x - z = -1.80 [/tex]
Now substitute x = -1.80 + z into one of the simplified equations. Let's use x - y + z = 2.20 :
[tex]\sf (-1.80 + z) - y + z = 2.20 [/tex]
[tex]\sf -1.80 + 2z - y = 2.20 [/tex]
[tex]\sf -y = 2.20 + 1.80 - 2z [/tex]
[tex]\sf -y = 4.00 - 2z [/tex]
[tex]\sf y = 2z - 4.00 [/tex]
Now substitute y = 2z - 4.00 and x = -1.80 + z back into one of the original equations to solve for z .
Let's use equation (1):
[tex]\sf 10(-1.80 + z) + 8(2z - 4.00) + 13z = 55.30 [/tex]
[tex]\sf -18 + 10z + 16z - 32 + 13z = 55.30 [/tex]
[tex]\sf 39z - 50 = 55.30 [/tex]
[tex]\sf 39z = 105.30 [/tex]
[tex]\sf z = \dfrac{105.30}{39} [/tex]
[tex]\sf z \approx 2.70 [/tex]
Now substitute z = 2.70 back into y = 2z - 4.00 and x = -1.80 + z to find y and x :
[tex]\sf y = 2(2.70) - 4.00 = 5.40 - 4.00 = 1.40 [/tex]
[tex]\sf x = -1.80 + 2.70 = 0.90 [/tex]
Therefore, the solution to the system is:
[tex]\sf x = 0.90, \quad y = 1.40, \quad z = 2.70 [/tex]
Hence, the cost of each type of fruit is:
- Cost of one apple ( x ): $0.90
- Cost of one cantaloupe ( y ): $1.40
- Cost of one bunch of bananas ( z ): $2.70
