To find the value of [tex]\( x \)[/tex] in the solution of the given system of linear equations, follow these steps:
1. Write down the given equations:
[tex]\[
\left\{
\begin{aligned}
6 + 4x - 2y &= 0 \quad \text{(1)} \\
-3 - 7y &= 10x \quad \text{(2)}
\end{aligned}
\right.
\][/tex]
2. Simplify the first equation:
First, we isolate one of the variables in Equation (1). Let's isolate [tex]\( y \)[/tex]:
[tex]\[
6 + 4x - 2y = 0
\][/tex]
[tex]\[
4x - 2y = -6
\][/tex]
3. Divide Equation (1) by 2 to further simplify:
[tex]\[
2x - y = -3 \quad \text{(3)}
\][/tex]
4. Rearrange Equation (2) for clarity:
[tex]\[
-3 - 7y = 10x
\][/tex]
[tex]\[
10x + 7y = -3 \quad \text{(4)}
\][/tex]
5. We now have a system of two equations in simplified form:
[tex]\[
\left\{
\begin{aligned}
2x - y &= -3 \quad \text{(3)} \\
10x + 7y &= -3 \quad \text{(4)}
\end{aligned}
\right.
\][/tex]
6. Use the elimination method to solve for one of the variables. We will eliminate [tex]\( y \)[/tex].
Multiply Equation (3) by 7 to prepare for elimination:
[tex]\[
7(2x - y) = 7(-3)
\][/tex]
[tex]\[
14x - 7y = -21 \quad \text{(5)}
\][/tex]
7. Subtract Equation (5) from Equation (4) to eliminate [tex]\( y \)[/tex]:
[tex]\[
(10x + 7y) - (14x - 7y) = -3 - (-21)
\][/tex]
[tex]\[
10x + 7y - 14x + 7y = -3 + 21
\][/tex]
[tex]\[
-4x + 14y = 18
\][/tex]
8. Solve for [tex]\( x \)[/tex]:
Converting the equation for [tex]\( x \)[/tex], we find:
[tex]\[
-4x + 14y = 18
\][/tex]
Simplifying by isolating [tex]\( 4x \)[/tex]:
[tex]\[
-4x = 18 - 14y
\][/tex]
Lastly, solving for [tex]\( x \)[/tex]:
[tex]\[
-4x = 18 - 14y \implies x = \frac{18 - 14y}{-4} = -1
\][/tex]
The value of [tex]\( x \)[/tex] in the solution of the system is:
[tex]\[ \boxed{-1} \][/tex]
