To solve the system of linear equations
[tex]\[
\begin{cases}
4x - 6y = -6 \\
8x + 3y = -7
\end{cases}
\][/tex]
we follow these steps:
1. Equation Rearrangement:
We already have the system in standard form [tex]\(Ax + By = C\)[/tex]. We can start solving it by any preferred method such as substitution or elimination.
2. Elimination Method:
We aim to eliminate one of the variables by adding or subtracting the equations. First, let's multiply the first equation by 2 to align the coefficients of [tex]\(x\)[/tex] in both equations.
[tex]\[
2 \times (4x - 6y) = 2 \times (-6)
\][/tex]
This simplifies to:
[tex]\[
8x - 12y = -12
\][/tex]
Now we have:
[tex]\[
\begin{cases}
8x - 12y = -12 \\
8x + 3y = -7
\end{cases}
\][/tex]
3. Subtract the Second Equation from the First:
[tex]\[
(8x - 12y) - (8x + 3y) = -12 - (-7)
\][/tex]
Simplify this:
[tex]\[
8x - 12y - 8x - 3y = -12 + 7
\][/tex]
[tex]\[
-15y = -5
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{-5}{-15} = \frac{1}{3}
\][/tex]
4. Substitute [tex]\(y\)[/tex] back into one of the original equations to solve for [tex]\(x\)[/tex]:
Let's use the first equation [tex]\(4x - 6y = -6\)[/tex]:
[tex]\[
4x - 6 \left(\frac{1}{3}\right) = -6
\][/tex]
Simplify the equation:
[tex]\[
4x - 2 = -6
\][/tex]
Add 2 to both sides:
[tex]\[
4x = -6 + 2
\][/tex]
[tex]\[
4x = -4
\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{-4}{4} = -1
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = \left(-1, \frac{1}{3}\right)
\][/tex]
