Certainly! Let's solve the system of linear equations using a step-by-step algebraic approach.
We are given the system of equations:
[tex]\[
\begin{cases}
x + 6y = -1 \\
2x - 2y = -7
\end{cases}
\][/tex]
Step 1: Solve one of the equations for one variable
Let's solve the first equation for [tex]\( x \)[/tex]:
[tex]\[ x + 6y = -1 \][/tex]
[tex]\[ x = -1 - 6y \][/tex]
Step 2: Substitute the expression for [tex]\( x \)[/tex] into the second equation
Substitute [tex]\( x = -1 - 6y \)[/tex] into the second equation [tex]\( 2x - 2y = -7 \)[/tex]:
[tex]\[ 2(-1 - 6y) - 2y = -7 \][/tex]
Step 3: Simplify and solve for [tex]\( y \)[/tex]
Distribute the 2:
[tex]\[ -2 - 12y - 2y = -7 \][/tex]
[tex]\[ -2 - 14y = -7 \][/tex]
Add 2 to both sides:
[tex]\[ -14y = -7 + 2 \][/tex]
[tex]\[ -14y = -5 \][/tex]
Divide both sides by -14:
[tex]\[ y = \frac{-5}{-14} \][/tex]
[tex]\[ y = \frac{5}{14} \][/tex]
Step 4: Substitute the value of [tex]\( y \)[/tex] back into the expression for [tex]\( x \)[/tex]
We already have [tex]\( x = -1 - 6y \)[/tex]. Now substitute [tex]\( y = \frac{5}{14} \)[/tex]:
[tex]\[ x = -1 - 6\left(\frac{5}{14}\right) \][/tex]
[tex]\[ x = -1 - \frac{30}{14} \][/tex]
[tex]\[ x = -1 - \frac{15}{7} \][/tex]
Convert -1 to a fraction with a denominator of 7:
[tex]\[ x = -\frac{7}{7} - \frac{15}{7} \][/tex]
[tex]\[ x = -\frac{22}{7} \][/tex]
Final answers:
[tex]\[
x = -\frac{22}{7} \quad \text{and} \quad y = \frac{5}{14}
\][/tex]
So the solution to the system of equations is:
[tex]\[
\left( x, y \right) = \left( -\frac{22}{7}, \frac{5}{14} \right)
\][/tex]
