Certainly! Let's solve the system of linear equations by substitution.
The given system of equations is:
[tex]\[
\begin{aligned}
-x - 2y &= -8 \quad \text{(1)} \\
y &= 6x - 9 \quad \text{(2)}
\end{aligned}
\][/tex]
We can use the substitution method as follows:
1. Solve one of the equations for one variable.
Equation (2) is already solved for [tex]\( y \)[/tex]:
[tex]\[
y = 6x - 9
\][/tex]
2. Substitute this expression into the other equation.
Substitute [tex]\( y = 6x - 9 \)[/tex] into Equation (1):
[tex]\[
-x - 2(6x - 9) = -8
\][/tex]
3. Simplify and solve for [tex]\( x \)[/tex].
[tex]\[
-x - 2(6x - 9) = -8
\][/tex]
Distribute the [tex]\(-2\)[/tex]:
[tex]\[
-x - 12x + 18 = -8
\][/tex]
Combine like terms:
[tex]\[
-13x + 18 = -8
\][/tex]
Subtract 18 from both sides:
[tex]\[
-13x = -8 - 18
\][/tex]
Simplify the right side:
[tex]\[
-13x = -26
\][/tex]
Divide by [tex]\(-13\)[/tex]:
[tex]\[
x = \frac{-26}{-13} = 2
\][/tex]
4. Substitute [tex]\( x \)[/tex] back into the expression for [tex]\( y \)[/tex].
[tex]\[
y = 6x - 9
\][/tex]
Substitute [tex]\( x = 2 \)[/tex]:
[tex]\[
y = 6(2) - 9 = 12 - 9 = 3
\][/tex]
So, the solution to the system of equations is:
[tex]\[
(x, y) = (2, 3)
\][/tex]
Verification:
To verify, substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex] back into the original equations:
For Equation (1):
[tex]\[
-x - 2y = -8
\][/tex]
[tex]\[
-(2) - 2(3) = -8
\][/tex]
[tex]\[
-2 - 6 = -8
\][/tex]
[tex]\[-8 = -8 \quad \text{(True)}\][/tex]
For Equation (2):
[tex]\[
y = 6x - 9
\][/tex]
\]
[tex]\[
3 = 6(2) - 9
=1]}
\][/tex]
\+9}
[tex]\[
3 = 12 - 9
\][/tex]
3
\]
{-9}
Both equations are satisfied with [tex]\( x = 2 \)[/tex] and [tex]\( y = 3 \)[/tex], confirming our solution is correct.
Thus, the solution to the system is:
[tex]\[
(x, y) = (2, 3)
\][/tex]
