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Solve the system of equations using the method of elimination.

[tex]\[
\begin{cases}
6x + 5y = 88 \\
5x + 6y = 88
\end{cases}
\][/tex]

[tex]\[
(x, y) = \boxed{\phantom{solution}}
\][/tex]

Sagot :

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].