To solve the system of linear equations:
[tex]\[
\begin{array}{l}
9x + 11y = -14 \\
6x - 5y = -34
\end{array}
\][/tex]
we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. Here's a step-by-step solution:
1. Label the equations:
[tex]\[
\begin{aligned}
&\text{(1)} \quad 9x + 11y = -14 \\
&\text{(2)} \quad 6x - 5y = -34
\end{aligned}
\][/tex]
2. Eliminate one variable:
To eliminate one of the variables, we need to make the coefficients of that variable in both equations the same (magnitude). Let's eliminate [tex]\(x\)[/tex]. To do this, we can find a common multiple of the coefficients of [tex]\(x\)[/tex] in both equations, which is [tex]\(54\)[/tex].
Multiply equation (1) by [tex]\(6\)[/tex] and equation (2) by [tex]\(9\)[/tex]:
[tex]\[
\begin{aligned}
&6(9x + 11y) = 6(-14) \\
&9(6x - 5y) = 9(-34)
\end{aligned}
\][/tex]
Simplifying these, we get:
[tex]\[
\begin{aligned}
&54x + 66y = -84 \quad \text{(3)} \\
&54x - 45y = -306 \quad \text{(4)}
\end{aligned}
\][/tex]
3. Subtract the new equations to eliminate [tex]\(x\)[/tex]:
Subtract equation (4) from equation (3):
[tex]\[
(54x + 66y) - (54x - 45y) = -84 - (-306)
\][/tex]
Simplifying this, we get:
[tex]\[
54x + 66y - 54x + 45y = -84 + 306
\][/tex]
[tex]\[
111y = 222
\][/tex]
4. Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{222}{111} = 2
\][/tex]
5. Substitute [tex]\(y\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex]:
Substitute [tex]\(y = 2\)[/tex] into equation (1):
[tex]\[
9x + 11(2) = -14
\][/tex]
Simplify and solve for [tex]\(x\)[/tex]:
[tex]\[
9x + 22 = -14
\][/tex]
[tex]\[
9x = -14 - 22
\][/tex]
[tex]\[
9x = -36
\][/tex]
[tex]\[
x = \frac{-36}{9} = -4
\][/tex]
6. Write the solution:
The solution to the system of equations is:
[tex]\[
x = -4, \quad y = 2
\][/tex]
Thus, the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations are [tex]\((-4, 2)\)[/tex].
