To solve the system of equations:
[tex]\[
\begin{cases}
2x + 7y = -1 \\
4x - 3y = -19
\end{cases}
\][/tex]
we will use the method of elimination. This involves making the coefficients of one of the variables the same (or opposites) so that we can eliminate that variable by adding or subtracting the equations.
Step 1: Make the coefficients of [tex]\(x\)[/tex] the same in both equations.
- The first equation is [tex]\(2x + 7y = -1\)[/tex].
- The second equation is [tex]\(4x - 3y = -19\)[/tex].
We can achieve this by multiplying the first equation by 2:
[tex]\[
2(2x + 7y) = 2(-1) \\
4x + 14y = -2
\][/tex]
So, our system now is:
[tex]\[
\begin{cases}
4x + 14y = -2 \\
4x - 3y = -19
\end{cases}
\][/tex]
Step 2: Subtract the second equation from the first equation to eliminate [tex]\(x\)[/tex]:
[tex]\[
(4x + 14y) - (4x - 3y) = -2 - (-19) \\
4x + 14y - 4x + 3y = -2 + 19 \\
17y = 17
\][/tex]
Step 3: Solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{17}{17} \\
y = 1
\][/tex]
Step 4: Substitute [tex]\(y = 1\)[/tex] into the first equation to solve for [tex]\(x\)[/tex]:
[tex]\[
2x + 7(1) = -1 \\
2x + 7 = -1 \\
2x = -1 - 7 \\
2x = -8 \\
x = \frac{-8}{2} \\
x = -4
\][/tex]
Thus, the solution to the system of equations is:
[tex]\[
(x, y) = (-4, 1)
\][/tex]
