Let's analyze the given system of equations to determine which statements are true. The equations are as follows:
- \( 2x - 7y = -13 \)
- \( -2x + 11y = 1 \)
To check the statements, we can display the system and perform the necessary calculations.
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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. -
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. -
The sum of the system of equations is \( 4y = -12 \).
After adding the two equations, we got \( 4y = -12 \). This statement is true. -
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. -
y = -3
From our previous calculations, we found that \( y = -3 \). This statement is true. -
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