Question

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

2 x minus 7 y = negative 13. Negative 2 x + 11 y = 1.
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 4 y = negative 12.
x = 17
y = negative 3
There are infinitely many solutions to the system of equations.

Answer 1

Let's analyze the given system of equations to determine which statements are true. The equations are as follows:

1) \( 2x - 7y = -13 \)
2) \( -2x + 11y = 1 \)

To check the statements, we can display the system and perform the necessary calculations.

1. **The x-variable will be eliminated when adding the system of equations.**
If we add the two equations, the \( 2x \) from the first equation and the \( -2x \) from the second equation will cancel out:
\[
(2x - 7y) + (-2x + 11y) = -13 + 1
\]
This simplifies to:
\[
0 + 4y = -12,
\]
so this statement is **true**.

2. **The y-variable will be eliminated when adding the system of equations.**
No, the \( y \)-variable will not be eliminated; instead, we will be left with an expression for \( y \). This statement is **false**.

3. **The sum of the system of equations is \( 4y = -12 \).**
After adding the two equations, we got \( 4y = -12 \). This statement is **true**.

4. **x = 17**
We will find \( x \) and \( y \) by solving the system. Starting from \( 4y = -12 \):
\[
y = -3.
\]
Substitute \( y = -3 \) back into one of the original equations to find \( x \):
\[
2x - 7(-3) = -13
\]
\[
2x + 21 = -13
\]
\[
2x = -34
\]
\[
x = -17.
\]
Thus, \( x \) is not 17; this statement is **false**.

5. **y = -3**
From our previous calculations, we found that \( y = -3 \). This statement is **true**.

6. **There are infinitely many solutions to the system of equations.**
Since we obtained a unique solution \( (x, y) = (-17, -3) \), there are not infinitely many solutions. This statement is **false**.

### Summary of True Statements:
- The x-variable will be eliminated when adding the system of equations.
- The sum of the system of equations is \( 4y = -12 \).
- y = -3