Question

Given the system of equations 2x – 8y = 0 and –2x + 4y = –8, which statements are true? Check all that apply.
The x-variable will cancel when adding the system of equations.
After adding the system of equations, you get –12y = –8.
y = 2
x = 8
The solution to the system is (2, 8).

Answer 1

Let's analyze the system of equations step by step:

1. The given equations are:
- Equation (1): \( 2x - 8y = 0 \)
- Equation (2): \( -2x + 4y = -8 \)

2. To see if the x-variable will cancel when adding the equations, let's add the two equations together:

\[
(2x - 8y) + (-2x + 4y) = 0 + (-8)
\]

This simplifies to:

\[
0x - 4y = -8 \implies -4y = -8
\]

We see that the x-variable does indeed cancel out.

3. Now solving for \( y \):

\[
-4y = -8 \implies y = \frac{-8}{-4} = 2
\]

4. Now substituting \( y = 2 \) back into one of the original equations to solve for \( x \). Using Equation (1):

\[
2x - 8(2) = 0 \implies 2x - 16 = 0 \implies 2x = 16 \implies x = \frac{16}{2} = 8
\]

Thus, \( x = 8 \).

5. So the solution to the system of equations is:

\(
(x, y) = (8, 2)
\)

Now, we can evaluate the statements provided:

- **The x-variable will cancel when adding the system of equations.**
- **True**. As we showed above, the x-variable does cancel.

- **After adding the system of equations, you get –12y = –8.**
- **False**. We found that after adding, we get \( -4y = -8 \).

- **y = 2**
- **True**. We calculated that \( y = 2 \).

- **x = 8**
- **True**. We found that \( x = 8 \).

- **The solution to the system is (2, 8).**
- **False**. The correct solution is \( (8, 2) \).

Based on this analysis, the true statements are:

1. The x-variable will cancel when adding the system of equations.
2. y = 2
3. x = 8