To solve the system of equations
[tex]\[
\left\{\begin{array}{l}
13x - 6y = 2 \\
3x - 4y = -10
\end{array}\right.
\][/tex]
we will use the method of elimination or substitution. Here is a detailed step-by-step solution:
1. Step 1: Multiply the second equation by a factor that will make the coefficients of either [tex]\(x\)[/tex] or [tex]\(y\)[/tex] in both equations the same (or opposites).
We can notice that multiplying the second equation by 2 will make the coefficient of [tex]\(y\)[/tex] in both equations equal (in magnitude):
[tex]\[
2(3x - 4y) = 2(-10)
\][/tex]
which simplifies to:
[tex]\[
6x - 8y = -20
\][/tex]
2. Step 2: Rewrite the system with the new equation:
[tex]\[
\left\{\begin{array}{l}
13x - 6y = 2 \\
6x - 8y = -20
\end{array}\right.
\][/tex]
3. Step 3: Eliminate one of the variables by combining the equations. To do this, we can multiply the first equation by 4 and the second equation by 3 to align the [tex]\(y\)[/tex] coefficients:
[tex]\[
4(13x - 6y) = 4(2) \implies 52x - 24y = 8
\][/tex]
[tex]\[
3(6x - 8y) = 3(-20) \implies 18x - 24y = -60
\][/tex]
4. Step 4: Subtract the second modified equation from the first modified equation to eliminate [tex]\(y\)[/tex]:
[tex]\[
(52x - 24y) - (18x - 24y) = 8 - (-60)
\][/tex]
This simplifies to:
[tex]\[
34x = 68
\][/tex]
5. Step 5: Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{68}{34} = 2
\][/tex]
6. Step 6: Substitute [tex]\(x = 2\)[/tex] back into one of the original equations to find the value of [tex]\(y\)[/tex]. We use the second original equation for this:
[tex]\[
3x - 4y = -10
\][/tex]
Substitute [tex]\(x = 2\)[/tex]:
[tex]\[
3(2) - 4y = -10
\][/tex]
[tex]\[
6 - 4y = -10
\][/tex]
[tex]\[
-4y = -10 - 6
\][/tex]
[tex]\[
-4y = -16
\][/tex]
7. Step 7: Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-16}{-4} = 4
\][/tex]
Therefore, the solution to the system of equations is [tex]\(x = 2\)[/tex] and [tex]\(y = 4\)[/tex].
The final answer is:
[tex]\[
\boxed{(2, 4)}
\][/tex]
