To solve the system of linear equations:
[tex]\[
\begin{cases}
-7x + 6y = 5 \\
7x + 2y = 11
\end{cases}
\][/tex]
we can use various methods such as substitution, elimination, or matrix methods. Here's a detailed, step-by-step solution using the elimination method:
1. Set Up the Equations:
[tex]\[
\begin{aligned}
\text{Equation 1:} & \quad -7x + 6y = 5 \\
\text{Equation 2:} & \quad 7x + 2y = 11
\end{aligned}
\][/tex]
2. Add the Equations:
First, observe that adding both equations will eliminate [tex]\( x \)[/tex] since the coefficients of [tex]\( x \)[/tex] are opposites ([tex]\(-7x\)[/tex] and [tex]\(7x\)[/tex]).
[tex]\[
(-7x + 6y) + (7x + 2y) = 5 + 11
\][/tex]
Simplifying this, we get:
[tex]\[
8y = 16
\][/tex]
3. Solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{16}{8}
\][/tex]
[tex]\[
y = 2
\][/tex]
4. Substitute [tex]\( y \)[/tex] back into One of the Original Equations:
Use Equation 2 to find [tex]\( x \)[/tex]:
[tex]\[
7x + 2(2) = 11
\][/tex]
Simplify the equation:
[tex]\[
7x + 4 = 11
\][/tex]
[tex]\[
7x = 11 - 4
\][/tex]
[tex]\[
7x = 7
\][/tex]
[tex]\[
x = \frac{7}{7}
\][/tex]
[tex]\[
x = 1
\][/tex]
5. Solution:
The solution to the system of equations is:
[tex]\[
x = 1 \quad \text{and} \quad y = 2
\][/tex]
Therefore, the point [tex]\( (x, y) \)[/tex] that satisfies both equations is [tex]\((1, 2)\)[/tex].
