To solve the simultaneous equations:
[tex]$
\left\{
\begin{array}{l}
x + 6y = -19 \\
5x + 4y = 9
\end{array}
\right.
$[/tex]
We can use the method of substitution or elimination. Let's use the elimination method. Here's a step-by-step solution:
1. Write the given equations clearly:
[tex]\[
\begin{aligned}
&(1)\quad x + 6y = -19 \\
&(2)\quad 5x + 4y = 9 \\
\end{aligned}
\][/tex]
2. Multiply Equation (1) by 5 to align the coefficient of \( x \) with Equation (2) for elimination:
[tex]\[
5(x + 6y) = 5(-19)
\][/tex]
Simplifying this gives us:
[tex]\[
5x + 30y = -95 \quad (3)
\][/tex]
3. Subtract Equation (2) from Equation (3) to eliminate \( x \):
[tex]\[
(5x + 30y) - (5x + 4y) = -95 - 9
\][/tex]
Simplifying the left-hand side (canceling \( 5x \)):
[tex]\[
30y - 4y = -104
\][/tex]
Which reduces to:
[tex]\[
26y = -104
\][/tex]
4. Solve for \( y \):
[tex]\[
y = \frac{-104}{26} = -4
\][/tex]
5. Substitute \( y = -4 \) into Equation (1) to find \( x \):
[tex]\[
x + 6(-4) = -19
\][/tex]
Simplifying this:
[tex]\[
x - 24 = -19
\][/tex]
So:
[tex]\[
x = -19 + 24
\][/tex]
Which gives:
[tex]\[
x = 5
\][/tex]
Therefore, the solution to the simultaneous equations is:
[tex]\[
x = 5 \quad \text{and} \quad y = -4
\][/tex]
