To solve the given system of equations:
[tex]\[
\begin{array}{c}
2x + 7y = -1 \\
4x - 3y = -19
\end{array}
\][/tex]
We can use the method of elimination to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Step 1: Align the equations and prepare them for elimination.
[tex]\[
\begin{array}{rl}
(1) & 2x + 7y = -1 \\
(2) & 4x - 3y = -19
\end{array}
\][/tex]
Step 2: Eliminate one variable.
We will eliminate [tex]\(x\)[/tex] by making the coefficients of [tex]\(x\)[/tex] in both equations equal. To do this, we can multiply equation [tex]\((1)\)[/tex] by 2:
[tex]\[
4x + 14y = -2 \quad \text{(which is our new equation (3))}
\][/tex]
Now, we have:
[tex]\[
\begin{array}{rl}
(3) & 4x + 14y = -2 \\
(2) & 4x - 3y = -19
\end{array}
\][/tex]
Step 3: Subtract equation (2) from equation (3) to eliminate [tex]\(x\)[/tex]:
[tex]\[
(4x + 14y) - (4x - 3y) = -2 - (-19)
\][/tex]
[tex]\[
4x + 14y - 4x + 3y = -2 + 19
\][/tex]
[tex]\[
17y = 17
\][/tex]
[tex]\[
y = 1
\][/tex]
Step 4: Substitute [tex]\(y = 1\)[/tex] back into one of the original equations to find [tex]\(x\)[/tex].
We substitute [tex]\(y = 1\)[/tex] into equation (1):
[tex]\[
2x + 7(1) = -1
\][/tex]
[tex]\[
2x + 7 = -1
\][/tex]
[tex]\[
2x = -1 - 7
\][/tex]
[tex]\[
2x = -8
\][/tex]
[tex]\[
x = -4
\][/tex]
Conclusion:
The solution to the system of equations is [tex]\((-4, 1)\)[/tex].
