Question

let the cost of one apple = x and one banana = y.
we can write the system of equations:

\\2x + y = 7\\
(from the first purchase)

\\x + 2y = 5\\
(from the second purchase)

Explanation:

Identify the system of equations

We are given a system of two linear equations representing purchases:

1. \(2x + y = 7\) (first purchase)
2. \(x + 2y = 5\) (second purchase)

Here, \(x\) represents the cost of one apple and \(y\) represents the cost of one banana.

Express one variable in terms of the other

From the first equation, we can isolate \(y\):

$$y = 7 - 2x$$

Substitute into the second equation

Substitute the expression for \(y\) into the second equation:

$$x + 2(7 - 2x) = 5$$

Expand and simplify the equation:

$$x + 14 - 4x = 5$$

$$-3x + 14 = 5$$

$$-3x = -9$$

$$x = 3$$

Solve for the remaining variable

Substitute \(x = 3\) back into the expression for \(y\):

$$y = 7 - 2(3)$$

$$y = 1$$

Verify the solution

Check the values in both original equations:

• First equation: \(2(3) + 1 = 7\) (True)
• Second equation: \(3 + 2(1) = 5\) (True)

Answer:

\(x = 3\), \(y = 1\)