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{6x-3y=18, -3x -9y = -93}

In order to solve this system of equations using the elimination/combination method, the first thing Emma should do is
Multiply the first equation by 3
Multiply the first equation by -2
Multiply the second equation by 2
Multiply the second equation by -3

3 answers

To use the elimination (or combination) method effectively, you typically want to manipulate the equations such that adding or subtracting them leads to the elimination of one of the variables.

Let's rewrite the system of equations:

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

Looking at the coefficients of \(x\) in both equations, we see that multiplying the second equation by 2 will simplify the comparison between the two. By doing this, we can eliminate the \(x\) variable effectively when we combine the equations.

Therefore, Emma should:

Multiply the second equation by 2.

If she does this, the second equation becomes:

\[ -6x - 18y = -186 \]

After that, combining with the first equation \( 6x - 3y = 18 \) would eliminate \(x\).

The value of x is

Let's solve the system of equations step-by-step:

The original equations are:

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

From the previous step, we decided to multiply the second equation by 2:

\[ -3x - 9y = -93 \quad \text{becomes} \quad -6x - 18y = -186 \]

Now we can write the system of equations as:

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

Next, we can add these two equations together to eliminate \(x\):

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

This simplifies to:

\[ -21y = -168 \]

Now, solving for \(y\):

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

Now that we have the value of \(y\), we can substitute it back into one of the original equations to find \(x\). We'll use the first equation:

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

This simplifies to:

\[ 6x - 24 = 18 \]

Now, adding 24 to both sides:

\[ 6x = 42 \]

Finally, dividing by 6:

\[ x = 7 \]

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