Answer :

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].