To determine the value of [tex]\( x \)[/tex] at which the graphs of the equations intersect, we need to solve the system of equations:
[tex]\[
\begin{cases}
2x - y = 6 \\
5x + 10y = -10
\end{cases}
\][/tex]
Let's go through the solution step by step:
1. Isolate one variable in one of the equations:
Start by isolating [tex]\( y \)[/tex] in the first equation:
[tex]\[
2x - y = 6
\][/tex]
Subtract [tex]\( 2x \)[/tex] from both sides:
[tex]\[
-y = 6 - 2x
\][/tex]
Then multiply by -1 to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 2x - 6
\][/tex]
2. Substitute the expression for [tex]\( y \)[/tex] into the other equation:
Now substitute [tex]\( y = 2x - 6 \)[/tex] into the second equation:
[tex]\[
5x + 10y = -10
\][/tex]
Replace [tex]\( y \)[/tex] with [tex]\( 2x - 6 \)[/tex]:
[tex]\[
5x + 10(2x - 6) = -10
\][/tex]
Simplify the equation:
[tex]\[
5x + 20x - 60 = -10
\][/tex]
Combine like terms:
[tex]\[
25x - 60 = -10
\][/tex]
Add 60 to both sides:
[tex]\[
25x = 50
\][/tex]
Divide by 25:
[tex]\[
x = 2
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] at which the graphs of the equations intersect is [tex]\( \boxed{2} \)[/tex].
