Consider the system of equations shown below to answer questions 20 and 21.

Question

6x − 3y = 18


−3x − 9y = −93


20. In order to solve this system of equations using the elimination/combination method, the first thing you should do is

(1 point)
Responses

multiply the first equation by 3
multiply the first equation by 3

multiply the first equation by -2
multiply the first equation by -2

multiply the second equation by 2
multiply the second equation by 2

multiply the second equation by -3

Answer 1

To use the elimination method on the given system of equations, we want to manipulate the equations so that when we add or subtract them, one of the variables will be eliminated.

The equations are:

1. \(6x - 3y = 18\)
2. \(-3x - 9y = -93\)

To eliminate \(y\), we can make the coefficients of \(y\) in both equations equal. The first equation has \(-3y\) and the second equation has \(-9y\). If we multiply the first equation by 3, we will have:

\[ 3(6x - 3y) = 3(18) \] Which simplifies to: \[ 18x - 9y = 54 \]

Now we can combine this with the second equation: \[ -3x - 9y = -93 \]

Now both equations will have \(-9y\), allowing us to eliminate \(y\) when we add them.

Thus, the correct choice is: multiply the first equation by 3.

Answer 2

The value of x will be?

(1 point)
Responses

-3
-3

7
7

8
8

18

Answer 3

To find the value of \( x \) after manipulating the given equations and using the elimination method, we continue from the equations that we derived previously.

Starting with:

1. \( 18x - 9y = 54 \) (after multiplying the first equation by 3)
2. \( -3x - 9y = -93 \) (the second equation unchanged)

Now we can add these two equations to eliminate \( y \):

\[ (18x - 9y) + (-3x - 9y) = 54 + (-93) \]

This simplifies to:

\[ (18x - 3x) + (-9y - 9y) = 54 - 93 \]

\[ 15x - 18y = -39 \]

But since we want to eliminate \( y \) directly, let's add the two modified equations:

\[ (18x - 9y) + (-3x - 9y) = 54 - 93 \]

This results in:

\[ 15x - 18y = -39 \]

Now, to isolate \( x \), we need to express \( y \) in terms of \( x \) using one of the original equations.

Let’s solve one of the original equations for \( y \):

From the original first equation: \[ 6x - 3y = 18 \]

Now, isolating \( y \): \[ -3y = 18 - 6x \] \[ y = \frac{6x - 18}{3} = 2x - 6 \]

Now substitute \( y \) into the second equation:

\[ -3x - 9(2x - 6) = -93 \]

Distributing: \[ -3x - 18x + 54 = -93 \] Combining like terms: \[ -21x + 54 = -93 \] Subtracting 54 from both sides: \[ -21x = -147 \] Dividing by -21: \[ x = 7 \]

Thus, the value of \( x \) is: 7.