Questions: Use the method of elimination to solve the following system of equations. If the system is dependent, express the solution set in terms of one of the variables. Leave all fractional answers in fraction form. 2x - 4y = 18 -10x + 20y = -91

Solution Steps

To solve the system of equations using the method of elimination, we first observe that the coefficients of \(x\) and \(y\) in both equations are multiples of each other. This suggests that the system might be dependent. We can multiply the first equation by 5 to make the coefficients of \(x\) in both equations equal in magnitude but opposite in sign. Then, we add the equations to eliminate \(x\) and check if the resulting equation is consistent. If it is consistent, the system is dependent, and we express the solution set in terms of one of the variables.

We are given the following system of equations:

\[ \begin{aligned}

1. & \quad 2x - 4y = 18 \\
2. & \quad -10x + 20y = -91 \end{aligned} \]

First, observe that the second equation is a multiple of the first equation. Let's simplify both equations to see if they are dependent.

Divide the second equation by \(-5\):

\[ -10x + 20y = -91 \quad \Rightarrow \quad 2x - 4y = \frac{91}{5} \]

Now, compare this with the first equation:

\[ 2x - 4y = 18 \]

The simplified form of the second equation, \(2x - 4y = \frac{91}{5}\), does not match the first equation, \(2x - 4y = 18\). This indicates that the two equations are not consistent with each other.

Final Answer