To solve the given system of linear equations:
[tex]\[
\begin{array}{l}
5x + 2y = 22 \\
-2x + 6y = 3
\end{array}
\][/tex]
we can use the substitution or elimination method. Here, the elimination method can be particularly effective. Follow these steps to find the solution:
1. First equation:
[tex]\[
5x + 2y = 22
\][/tex]
2. Second equation:
[tex]\[
-2x + 6y = 3
\][/tex]
For elimination, align the equations to eliminate one of the variables.
3. Multiply the first equation by 3 to align the coefficients of [tex]\( y \)[/tex] in both equations:
[tex]\[
3(5x + 2y) = 3(22)
\][/tex]
This simplifies to:
[tex]\[
15x + 6y = 66
\][/tex]
4. Now, we have:
[tex]\[
\begin{array}{l}
15x + 6y = 66 \\
-2x + 6y = 3
\end{array}
\][/tex]
5. Eliminate [tex]\( y \)[/tex] by subtracting the second equation from the first:
[tex]\[
(15x + 6y) - (-2x + 6y) = 66 - 3
\][/tex]
Simplifying this, we get:
[tex]\[
15x + 6y + 2x - 6y = 63
\][/tex]
[tex]\[
17x = 63
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{63}{17}
\][/tex]
7. Convert this to a decimal and round to the nearest tenth:
[tex]\[
x \approx 3.7
\][/tex]
So, the [tex]\( x \)[/tex]-coordinate of the solution is:
[tex]\[
\boxed{3.7}
\][/tex]
