To solve the system of linear equations:
[tex]\[
\begin{array}{l}
2x + 7y = 4 \\
-4x - 3y = 14
\end{array}
\][/tex]
we follow these steps:
1. Multiply the first equation by 2 to make the coefficients of [tex]\( x \)[/tex] in the two equations opposites of each other:
[tex]\[
2(2x + 7y) = 2(4)
\][/tex]
This yields:
[tex]\[
4x + 14y = 8
\][/tex]
2. Now, we have the modified system of equations:
[tex]\[
\begin{array}{l}
4x + 14y = 8 \\
-4x - 3y = 14
\end{array}
\][/tex]
3. Add the two equations to eliminate [tex]\( x \)[/tex]:
[tex]\[
(4x + 14y) + (-4x - 3y) = 8 + 14
\][/tex]
Simplifying the left side and summing the right side, we obtain:
[tex]\[
4x - 4x + 14y - 3y = 22 \\
11y = 22
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{22}{11} \\
y = 2
\][/tex]
5. Substitute the value of [tex]\( y \)[/tex] back into the first equation to solve for [tex]\( x \)[/tex]:
[tex]\[
2x + 7(2) = 4 \\
2x + 14 = 4 \\
2x = 4 - 14 \\
2x = -10 \\
x = \frac{-10}{2} \\
x = -5
\][/tex]
Therefore, the solution to the system of equations is the ordered pair:
[tex]\[
(-5,2)
\][/tex]
