Let's solve the given system of linear equations step by step:
[tex]\[
\left\{
\begin{array}{c}
\frac{2}{3} x - \frac{3}{4} y = \frac{1}{6} \\
\frac{1}{8} x - \frac{5}{6} y = 12
\end{array}
\right.
\][/tex]
Let's label them as Equation 1 and Equation 2:
Equation 1:
[tex]\[
\frac{2}{3} x - \frac{3}{4} y = \frac{1}{6}
\][/tex]
Equation 2:
[tex]\[
\frac{1}{8} x - \frac{5}{6} y = 12
\][/tex]
Step 1: Clear fractions by finding the common denominators.
For Equation 1, the common denominator of 3, 4, and 6 is 12, so we multiply every term by 12:
[tex]\[
12 \left(\frac{2}{3} x\right) - 12 \left(\frac{3}{4} y\right) = 12 \left(\frac{1}{6}\right)
\][/tex]
This simplifies to:
[tex]\[
8x - 9y = 2
\][/tex]
For Equation 2, the common denominator of 8 and 6 is 24, so we multiply every term by 24:
[tex]\[
24 \left(\frac{1}{8} x\right) - 24 \left(\frac{5}{6} y\right) = 24 \cdot 12
\][/tex]
This simplifies to:
[tex]\[
3x - 20y = 288
\][/tex]
Now we have the system:
[tex]\[
\left\{
\begin{array}{c}
8x - 9y = 2 \\
3x - 20y = 288
\end{array}
\right.
\][/tex]
Step 2: Use the method of substitution or elimination to solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. Here we'll use the elimination method.
First, we manipulate the equations for easier elimination. Multiply Equation 1 by 3 and Equation 2 by 8:
[tex]\[
3(8x - 9y) = 3 \cdot 2
\][/tex]
[tex]\[
8(3x - 20y) = 8 \cdot 288
\][/tex]
This gives us:
[tex]\[
24x - 27y = 6
\][/tex]
[tex]\[
24x - 160y = 2304
\][/tex]
Now subtract the first modified equation from the second one:
[tex]\[
(24x - 160y) - (24x - 27y) = 2304 - 6
\][/tex]
This reduces to:
[tex]\[
-133y = 2298
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{2298}{-133} \approx -17.278
\][/tex]
Step 3: Substitute [tex]\(y\)[/tex] back into one of the original simplified equations to find [tex]\(x\)[/tex]. We use the first simplified equation:
[tex]\[
8x - 9(-17.278) = 2
\][/tex]
This simplifies to:
[tex]\[
8x + 155.502 = 2
\][/tex]
Subtract 155.502 from both sides:
[tex]\[
8x = 2 - 155.502
\][/tex]
[tex]\[
8x = -153.502
\][/tex]
Dividing by 8:
[tex]\[
x = \frac{-153.502}{8} \approx -19.188
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x \approx -19.188
\][/tex]
[tex]\[
y \approx -17.278
\][/tex]
