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/combination method effectively, we want to align the coefficients of one of the variables so that they can cancel out when we add or subtract the equations.

Looking at the equations:

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

We can notice that the coefficient of \(x\) in the second equation is \(-3\), and we can multiply the first equation by \(-1\) to change that coefficient but since that's not an option, we can choose another route.

Instead, multiplying the second equation by \(2\) will give us:

\(-6x - 18y = -186\)

This means we will then have:

1. \(6x - 3y = 18\) (the first equation)
2. \(-6x - 18y = -186\) (the second equation multiplied by 2)

Now, by adding these two equations together, we can eliminate the \(x\) variable.

Thus, the best option to eliminate variable \(x\) using the elimination method is to:

multiply the second equation by 2.

Answer 2

what will be the value of x?

Answer 3

To find the value of \( x \), we can solve the system of equations given:

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

From the earlier step, we determined it was effective to multiply Equation 2 by 2 to align and eliminate \( x \). Let's see how we can solve the system step-by-step.

First, we multiply Equation 2 by 2:

\[ -3x - 9y = -93 \implies -6x - 18y = -186 \quad \text{(Equation 2 multiplied by 2)} \]

Now we can write the system:

1. \( 6x - 3y = 18 \) (Equation 1)
2. \( -6x - 18y = -186 \) (Modified Equation 2)

Now, let's add Equation 1 and the modified Equation 2 together:

\[ (6x - 3y) + (-6x - 18y) = 18 - 186 \]

This simplifies to:

\[ 0x - 21y = -168 \]

So we have:

\[ -21y = -168 \]

Dividing both sides by \(-21\):

\[ y = \frac{-168}{-21} \implies y = 8 \]

Now that we have \( y \), we can substitute it back into one of the original equations to solve for \( x \). We'll use Equation 1:

\[ 6x - 3y = 18 \]

Substituting \( y = 8 \):

\[ 6x - 3(8) = 18 \] \[ 6x - 24 = 18 \]

Now, add 24 to both sides:

\[ 6x = 18 + 24 \] \[ 6x = 42 \]

Now divide by 6:

\[ x = \frac{42}{6} \implies x = 7 \]

So, the value of \( x \) is 7.