Question

Consider the system of equations.

3x + 2y = 16,

x + y = 4

Which numerical values could you multiply the second equation by to eliminate a variable? Select all that apply.
2
–2
3
–3
4
–4

Answer 1

To eliminate a variable when solving the system of equations, we can multiply the second equation by a numerical value so that the coefficients of one of the variables in both equations become opposites.

The given equations are:

1. \(3x + 2y = 16\) (Equation 1)
2. \(x + y = 4\) (Equation 2)

### To eliminate \(y\):

1. The coefficient of \(y\) in Equation 1 is \(2\).
2. To eliminate \(y\), we want the coefficient of \(y\) in Equation 2 to become \(-2\).

Thus, we can multiply the whole of Equation 2 by \(-2\):

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

Now we have the equations:

1. \(3x + 2y = 16\)
2. \(-2x - 2y = -8\)

Now the \(2y\) from Equation 1 and \(-2y\) from Equation 2 can be eliminated.

### To eliminate \(x\):

1. The coefficient of \(x\) in Equation 1 is \(3\).
2. To eliminate \(x\), we want the coefficient of \(x\) in Equation 2 to become \(-3\).

Thus, we can multiply the whole of Equation 2 by \(-3\):

\[
-3(x + y) = -3(4) \implies -3x - 3y = -12
\]

Now we have the equations:

1. \(3x + 2y = 16\)
2. \(-3x - 3y = -12\)

Now the \(3x\) from Equation 1 and \(-3x\) from Equation 2 can be eliminated.

### Summary:

To eliminate a variable (either \(y\) or \(x\)), you can multiply the second equation by:

- \(-2\) (to eliminate \(y\))
- \(-3\) (to eliminate \(x\))

**Thus, the correct values are:**

- \(-2\)
- \(-3\)

The other options (2, 3, 4, and -4) will not effectively eliminate a variable in this particular system of equations.