To find the coordinates of the center of the circle with given diameters, we need to determine the intersection point of the two lines represented by the equations of the diameters. Let's break this down step-by-step:
1. Write down the equations of the two diameters:
[tex]\[
\begin{align*}
x + y &= 5 \quad \text{(Equation 1)} \\
2x - y &= 1 \quad \text{(Equation 2)}
\end{align*}
\][/tex]
2. Solve the system of linear equations to find the intersection point:
From Equation 1:
[tex]\[
y = 5 - x
\][/tex]
Substitute [tex]\( y = 5 - x \)[/tex] into Equation 2:
[tex]\[
2x - (5 - x) = 1
\][/tex]
3. Simplify the equation:
[tex]\[
2x - 5 + x = 1
\][/tex]
[tex]\[
3x - 5 = 1
\][/tex]
[tex]\[
3x = 6
\][/tex]
[tex]\[
x = 2
\][/tex]
4. Substitute [tex]\( x = 2 \)[/tex] back into the expression for [tex]\( y \)[/tex] from Equation 1:
[tex]\[
y = 5 - x = 5 - 2 = 3
\][/tex]
Thus, the coordinates of the center of the circle, which is the intersection point of the two diameters, are:
[tex]\[
(x, y) = (2, 3)
\][/tex]
