To solve the system of equations
[tex]\[
\begin{cases}
6x + 5y = 88 \\
5x + 6y = 88
\end{cases}
\][/tex]
we'll use the method of elimination. Here's a step-by-step solution:
1. Label the equations:
[tex]\[
\begin{aligned}
\text{Equation 1:} & \quad 6x + 5y = 88 \\
\text{Equation 2:} & \quad 5x + 6y = 88
\end{aligned}
\][/tex]
2. Multiply Equation 1 by 5 and Equation 2 by 6 to make the coefficients of [tex]\(x\)[/tex] in both equations the same:
[tex]\[
\begin{aligned}
30x + 25y &= 440 \quad \text{(Equation 1 multiplied by 5)} \\
30x + 36y &= 528 \quad \text{(Equation 2 multiplied by 6)}
\end{aligned}
\][/tex]
3. Subtract the first modified equation from the second modified equation to eliminate [tex]\(x\)[/tex]:
[tex]\[
(30x + 36y) - (30x + 25y) = 528 - 440
\][/tex]
Simplifying this, we get:
[tex]\[
30x + 36y - 30x - 25y = 88
\][/tex]
Simplifying further:
[tex]\[
11y = 88
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{88}{11} = 8
\][/tex]
5. Substitute [tex]\(y = 8\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
Using Equation 1:
[tex]\[
6x + 5(8) = 88
\][/tex]
Simplifying:
[tex]\[
6x + 40 = 88
\][/tex]
[tex]\[
6x = 88 - 40
\][/tex]
[tex]\[
6x = 48
\][/tex]
[tex]\[
x = \frac{48}{6} = 8
\][/tex]
6. Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (8, 8)
\][/tex]
So, the solution to the system of equations is [tex]\((8, 8)\)[/tex].
