To solve the given system of equations and check the validity of the statements, let's proceed step-by-step.
The system of equations is:
[tex]\[
\begin{align*}
1) & \quad 2x - 8y = 0 \\
2) & \quad -2x + 4y = -8
\end{align*}
\][/tex]
### Step 1: Add the Equations
First, let's add the two equations to see if the [tex]\( x \)[/tex]-variable will cancel out.
[tex]\[
(2x - 8y) + (-2x + 4y) = 0 + (-8)
\][/tex]
[tex]\[
2x - 8y - 2x + 4y = -8
\][/tex]
This simplifies to:
[tex]\[
-4y = -8
\][/tex]
So, after adding the equations, we get:
[tex]\[
-4y = -8
\][/tex]
### Step 2: Solve for [tex]\( y \)[/tex]
Now, solve for [tex]\( y \)[/tex]:
[tex]\[
-4y = -8
\][/tex]
Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
y = 2
\][/tex]
Thus, [tex]\( y = 2 \)[/tex].
### Step 3: Solve for [tex]\( x \)[/tex]
Next, substitute [tex]\( y = 2 \)[/tex] back into one of the original equations to find [tex]\( x \)[/tex]. We'll use the first equation:
[tex]\[
2x - 8(2) = 0
\][/tex]
[tex]\[
2x - 16 = 0
\][/tex]
Add 16 to both sides:
[tex]\[
2x = 16
\][/tex]
Divide both sides by 2:
[tex]\[
x = 8
\][/tex]
Thus, [tex]\( x = 8 \)[/tex].
### Step 4: Solution to the System
The solution to the system of equations is [tex]\( (x, y) = (8, 2) \)[/tex].
### Step 5: Check the Statements
Now, let's verify the given statements based on our findings:
1. The [tex]\( x \)[/tex]-variable will cancel when adding the system of equations.
- This is True. When we added the equations, the terms involving [tex]\( x \)[/tex] canceled out.
2. After adding the system of equations, you get [tex]\( -12y = -8 \)[/tex].
- This is False. After adding the equations, we actually got [tex]\( -4y = -8 \)[/tex].
3. [tex]\( y = 2 \)[/tex].
- This is True. We found [tex]\( y = 2 \)[/tex] after solving the simplified equation.
4. [tex]\( x = 8 \)[/tex].
- This is True. We found [tex]\( x = 8 \)[/tex] from substituting [tex]\( y = 2 \)[/tex] into one of the original equations.
5. The solution to the system is [tex]\( (2, 8) \)[/tex].
- This is False. The correct solution to the system is [tex]\( (8, 2) \)[/tex].
Thus, the correct statements are:
- The [tex]\( x \)[/tex]-variable will cancel when adding the system of equations.
- [tex]\( y = 2 \)[/tex]
- [tex]\( x = 8 \)[/tex]
