Question

Tatum wants to solve the following systems of equations by using the elimination method. What integer should she multiply the second equation by to eliminate a variable when you add the equations?
6x+3y=14
3x+2y=10

Answer 1

The answers are 2, 6, -6, or -1/2

Answer 2

To use the elimination method to solve the system of equations:

1. \( 6x + 3y = 14 \) (Equation 1)
2. \( 3x + 2y = 10 \) (Equation 2)

we want to eliminate one of the variables by making the coefficients of either \( x \) or \( y \) the same in both equations.

Let's eliminate \( y \):

The coefficient of \( y \) in Equation 1 is \( 3 \), and the coefficient of \( y \) in Equation 2 is \( 2 \). To eliminate \( y \), we can multiply the second equation by \( 1.5 \) (or \( \frac{3}{2} \)). This way, the coefficient of \( y \) in Equation 2 will become \( 3 \), matching that of Equation 1.

Let's multiply Equation 2 by \( 1.5 \):

\[
1.5(3x + 2y) = 1.5(10)
\]

This simplifies to:

\[
4.5x + 3y = 15
\]

Now, the system of equations becomes:

1. \( 6x + 3y = 14 \)
2. \( 4.5x + 3y = 15 \)

To eliminate \( y \), we can subtract the two equations:

\[
(6x + 3y) - (4.5x + 3y) = 14 - 15
\]

This results in:

\[
1.5x = -1
\]

Solving for \( x \):

\[
x = \frac{-1}{1.5} = -\frac{2}{3}
\]

If you are required to eliminate \( y \), you can multiply the **second equation** \( 3x + 2y = 10 \) by \( 3 \):

When you multiply the second equation by \( 3 \):

\[
3(3x + 2y = 10)
\]
gives:
\[
9x + 6y = 30
\]

Now we can proceed to add/subtract the equations accordingly.

The integer Tatum should multiply the second equation by to eliminate \( y \) when you add the equations is:

\[
\text{multiply } 3 \text{ to the second equation.}
\]