Question

Which statements about this system of equations are true? Check all that apply.

Negative x + 6 y = 16. 8 x minus 6 y = negative 2.
The x-variable will be eliminated when adding the system of equations.
The y-variable will be eliminated when adding the system of equations.
The sum of the system of equations is Negative 8 x = 14.
x = 2
y = 3
There is only one solution to the system of equations.

Answer 1

To analyze the given system of equations and verify the statements, let's first rewrite the equations in a more standard form:

1. \(-x + 6y = 16\) (Equation 1)
2. \(8x - 6y = -2\) (Equation 2)

### Checking each statement:

1. **The x-variable will be eliminated when adding the system of equations.**
- To check this, let's add the two equations. We can first rewrite Equation 1 in a form that makes it easier to add:
\[
-x + 6y + 8x - 6y = 16 - 2
\]
Simplifying this gives:
\[
7x = 14 \quad (\text{which means the x-variable is not eliminated})
\]
- **This statement is false.**

2. **The y-variable will be eliminated when adding the system of equations.**
- From the same setup as above where we added the equations, we can see that the \(6y\) and \(-6y\) will cancel each other out:
\[
-x + 6y + 8x - 6y = 16 - 2
\]
This simplifies to:
\[
7x = 14 \quad (\text{the y-variable is eliminated})
\]
- **This statement is true.**

3. **The sum of the system of equations is Negative 8 x = 14.**
- From the previous addition, we found \(7x = 14\), not \(-8x\).
- **This statement is false.**

4. **x = 2.**
- Solving \(7x = 14\):
\[
x = \frac{14}{7} = 2
\]
- Hence, **this statement is true.**

5. **y = 3.**
- Now we can substitute \(x = 2\) back into one of the original equations to find \(y\). Using Equation 1:
\[
-2 + 6y = 16 \rightarrow 6y = 18 \rightarrow y = 3
\]
- Hence, **this statement is true.**

6. **There is only one solution to the system of equations.**
- Since we found a single solution \((x, y) = (2, 3)\) and the system appears to be consistent and independent (the equations represent two lines that intersect at one point), we conclude there is one solution.
- **This statement is true.**

### Summary of True Statements:
- The y-variable will be eliminated when adding the system of equations.
- x = 2.
- y = 3.
- There is only one solution to the system of equations.