Question

1. -2x + 12y = 18

4x - 2y = 30

Explanation:

We have a system of linear equations:
\[

$$\begin{cases} -2x + 12y = 18 \\ 4x - 2y = 30 \end{cases}$$

\]
We can use the elimination method. First, let's multiply the first equation by 2 to make the coefficients of \(x\) in both equations opposites (or to eliminate \(x\)).

Step 1: Multiply the first equation by 2

Multiply each term in \(-2x + 12y = 18\) by 2:
\[
2\times(-2x)+2\times(12y)=2\times18
\]
\[
-4x + 24y = 36
\]

Step 2: Add the new first equation and the second equation

Now we have the two equations:
\[
-4x + 24y = 36
\]
\[
4x - 2y = 30
\]
Add them together:
\[
(-4x + 4x)+(24y - 2y)=36 + 30
\]
Simplify:
\[
22y = 66
\]

Step 3: Solve for \(y\)

Divide both sides of \(22y = 66\) by 22:
\[
y=\frac{66}{22}=3
\]

Step 4: Substitute \(y = 3\) into one of the original equations to solve for \(x\)

Let's use the second equation \(4x - 2y = 30\). Substitute \(y = 3\):
\[
4x - 2\times3 = 30
\]
\[
4x - 6 = 30
\]
Add 6 to both sides:
\[
4x=30 + 6
\]
\[
4x=36
\]
Divide both sides by 4:
\[
x=\frac{36}{4}=9
\]

Answer:

The solution to the system of equations is \(x = 9\) and \(y = 3\), or as an ordered pair \((9, 3)\).