To solve the given system of equations:
[tex]\[
\begin{cases}
2x + 7y = -7 \\
-4x - 3y = -19
\end{cases}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that simultaneously satisfy both equations. Here’s a detailed step-by-step process:
1. Label the equations:
[tex]\[
\begin{aligned}
&\text{(1)} \quad 2x + 7y = -7 \\
&\text{(2)} \quad -4x - 3y = -19
\end{aligned}
\][/tex]
2. Multiply the equations to align coefficients for elimination:
We'll multiply the first equation by 2 to match the coefficients of [tex]\( x \)[/tex] in magnitude with that in the second equation:
[tex]\[
2 \times (2x + 7y) = 2 \times (-7)
\][/tex]
This gives us:
[tex]\[
4x + 14y = -14 \quad \text{(3)}
\][/tex]
3. Add equation (3) to equation (2):
[tex]\[
(4x + 14y) + (-4x - 3y) = -14 + (-19)
\][/tex]
Simplify the left and right sides:
[tex]\[
(4x - 4x) + (14y - 3y) = -14 - 19
\][/tex]
This simplifies to:
[tex]\[
11y = -33
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-33}{11} = -3
\][/tex]
5. Substitute [tex]\( y = -3 \)[/tex] back into equation (1) to find [tex]\( x \)[/tex]:
[tex]\[
2x + 7(-3) = -7
\][/tex]
Simplify:
[tex]\[
2x - 21 = -7
\][/tex]
Add 21 to both sides:
[tex]\[
2x = 14
\][/tex]
Divide by 2:
[tex]\[
x = 7
\][/tex]
The solution to the system of equations is [tex]\( x = 7 \)[/tex] and [tex]\( y = -3 \)[/tex]. Expressed as an ordered pair, the solution is:
[tex]\[
(7,-3)
\][/tex]
