Solve the system by substitution. If the system is inconsistent or has dependent equations, state so.

[tex]\[
\begin{aligned}
-x - 9y &= -19 \\
y &= 6x - 4
\end{aligned}
\][/tex]



Answer :

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].