Question

a system of equations is shown.
y = \frac{1}{2}x + 2
y = -x + 5
which ordered pair represents a solution to
the system of equations?
\bigcirc (1,4)
\bigcirc (2,3)
\bigcirc (0,2)
\bigcirc (3,0)

Explanation:

Step1: Set the equations equal

Since both equal \( y \), set \( \frac{1}{2}x + 2 = -x + 5 \).

Step2: Solve for \( x \)

Add \( x \) to both sides: \( \frac{1}{2}x + x + 2 = 5 \) → \( \frac{3}{2}x + 2 = 5 \).
Subtract 2: \( \frac{3}{2}x = 3 \).
Multiply by \( \frac{2}{3} \): \( x = 3 \times \frac{2}{3} = 2 \).

Step3: Find \( y \)

Substitute \( x = 2 \) into \( y = -x + 5 \): \( y = -2 + 5 = 3 \).
So the solution is \( (2, 3) \). Check with \( y = \frac{1}{2}x + 2 \): \( \frac{1}{2}(2) + 2 = 1 + 2 = 3 \), which matches.

Answer:

B. (2,3)