https://file.aiquickdraw.com/mathbot/file/1726016550392e323f4pz.jpg

Let's solve the system of equations shown in the image:

The system of equations is:

\[
\begin{aligned}
-2x + y &= 4 \\
4x - 6y &= -48
\end{aligned}
\]

### Step 1: Eliminate one variable

We can eliminate one variable by multiplying the first equation so that the coefficients of \(x\) or \(y\) in both equations become equal or opposites. Let’s eliminate \(x\) by multiplying the first equation by 2:

\[
2(-2x + y = 4) \quad \Rightarrow \quad -4x + 2y = 8
\]

Now we have the system:

\[
\begin{aligned}
-4x + 2y &= 8 \\
4x - 6y &= -48
\end{aligned}
\]

### Step 2: Add the equations

Now, add the two equations to eliminate \(x\):

\[
(-4x + 2y) + (4x - 6y) = 8 + (-48)
\]

This simplifies to:

\[
0x - 4y = -40
\]

Or:

\[
-4y = -40
\]

### Step 3: Solve for \(y\)

Now solve for \(y\):

\[
y = \frac{-40}{-4} = 10
\]

### Step 4: Substitute \(y\) back into one of the original equations

Substitute \(y = 10\) into the first equation:

\[
-2x + y = 4 \quad \Rightarrow \quad -2x + 10 = 4
\]

Solve for \(x\):

\[
-2x = 4 - 10 \quad \Rightarrow \quad -2x = -6 \quad \Rightarrow \quad x = 3
\]

### Final Answer:

The solution to the system is \(x = 3\) and \(y = 10\).

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. How would you graphically interpret the solution to this system of equations?
2. Can you solve the system using substitution instead of elimination?
3. What happens if both equations are parallel? How can you identify that?
4. How does changing the coefficients affect the method of solving?
5. Can you solve a system of three variables using a similar elimination method?

**Tip:** Always double-check your solution by substituting it back into the original equations to ensure it's correct.

Solve the system of equations -2x + y = 4 and 4x - 6y = -48 by combining the equations.

Learn how to solve a system of linear equations using the elimination method. This example demonstrates solving -2x + y = 4 and 4x - 6y = -48 step by step. Suitable for students in Grades 8-10.

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