To solve the system of equations using the elimination method, let's first write the equations:
- \( 6x - 3y = 18 \) (Equation 1)
- \( -3x - 9y = -93 \) (Equation 2)
Looking at the coefficients of \(y\) in both equations, we can manipulate these equations in a way that allows for elimination. The coefficients of \(y\) are -3 in Equation 1 and -9 in Equation 2.
To eliminate \(y\), we can multiply Equation 1 by 3, which will give us:
\[ 3(6x - 3y) = 3(18) \implies 18x - 9y = 54 \quad \text{(Modified Equation 1)} \]
Now, we have:
- \( 18x - 9y = 54 \) (Modified Equation 1)
- \( -3x - 9y = -93 \) (Equation 2)
Now, we can eliminate \(y\) by subtracting Equation 2 from Modified Equation 1:
\[ (18x - 9y) - (-3x - 9y) = 54 - (-93) \] \[ 18x - 9y + 3x + 9y = 54 + 93 \] \[ 21x = 147 \]
Now, divide by 21:
\[ x = \frac{147}{21} = 7 \]
So the value of \(x\) is \(7\).
Keep in mind that we could also choose another operation that would facilitate elimination; the given options might lead to the correct answer as long as they are chosen wisely. Thus, based on the options provided in the original question, the most effective first decision for Emma would be to multiply the first equation by 3.
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