To solve the system of equations:
[tex]\[
\begin{cases}
4x + 2y = 18 \\
-6y = 4x + 2
\end{cases}
\][/tex]
Follow these steps:
1. Rewrite the second equation to isolate one of the variables. Let's isolate [tex]\( x \)[/tex] in the second equation:
[tex]\[
-6y = 4x + 2
\][/tex]
We can rewrite this as:
[tex]\[
4x = -6y - 2
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-6y - 2}{4}
\][/tex]
Simplify the equation:
[tex]\[
x = \frac{-6y}{4} - \frac{2}{4}
\][/tex]
[tex]\[
x = \frac{-3y}{2} - \frac{1}{2}
\][/tex]
2. Substitute [tex]\( x \)[/tex] from the second equation into the first equation.
[tex]\[
4 \left( \frac{-3y}{2} - \frac{1}{2} \right) + 2y = 18
\][/tex]
Distribute the 4:
[tex]\[
4 \cdot \frac{-3y}{2} + 4 \cdot \frac{-1}{2} + 2y = 18
\][/tex]
[tex]\[
2(-3y) + 2(-1) + 2y = 18
\][/tex]
[tex]\[
-6y - 2 + 2y = 18
\][/tex]
Combine like terms:
[tex]\[
-6y + 2y = 18 + 2
\][/tex]
[tex]\[
-4y = 20
\][/tex]
Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{20}{-4}
\][/tex]
[tex]\[
y = -5
\][/tex]
3. Substitute [tex]\( y \)[/tex] back into the [tex]\( x \)[/tex]-isolation equation to find [tex]\( x \)[/tex].
[tex]\[
x = \frac{-3(-5)}{2} - \frac{1}{2}
\][/tex]
[tex]\[
x = \frac{15}{2} - \frac{1}{2}
\][/tex]
Subtract the fractions:
[tex]\[
x = \frac{15 - 1}{2}
\][/tex]
[tex]\[
x = \frac{14}{2}
\][/tex]
[tex]\[
x = 7
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
x = 7, \quad y = -5
\][/tex]
