To solve the system of equations by substitution, we'll follow these steps:
Given the system:
[tex]\[
\begin{aligned}
-x - 9y &= -19 \\
y &= 6x - 4
\end{aligned}
\][/tex]
1. Substitute the expression for [tex]\( y \)[/tex] from the second equation into the first equation:
From the second equation, we have:
[tex]\[ y = 6x - 4 \][/tex]
Now, substitute this value into the first equation:
[tex]\[
-x - 9(6x - 4) = -19
\][/tex]
2. Simplify the equation:
Distribute the [tex]\(-9\)[/tex] through the terms inside the parentheses:
[tex]\[
-x - 54x + 36 = -19
\][/tex]
Combine like terms:
[tex]\[
-55x + 36 = -19
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Move the constant term to the other side by subtracting 36 from both sides:
[tex]\[
-55x = -19 - 36
\][/tex]
Which simplifies to:
[tex]\[
-55x = -55
\][/tex]
Divide both sides by [tex]\(-55\)[/tex]:
[tex]\[
x = 1
\][/tex]
4. Substitute [tex]\( x = 1 \)[/tex] back into the second equation to find [tex]\( y \)[/tex]:
Use the expression for [tex]\( y \)[/tex] from the second equation:
[tex]\[
y = 6(1) - 4
\][/tex]
Simplify:
[tex]\[
y = 6 - 4
\][/tex]
[tex]\[
y = 2
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[
x = 1 \quad \text{and} \quad y = 2
\][/tex]
These values satisfy both original equations, so the system is consistent and the solution is [tex]\((1, 2)\)[/tex].
