Questions: Use the method of substitution 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. -4x + 4y = 20 2x = -42

$Use the method of substitution to solve the following system of equations. If the system is dependent, express the solution set in terms of one of the variables. Leave all fractional answers in fraction form. -4x + 4y = 20 2x = -42$

Transcript text: Use the method of substitution 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. \[ \left\{\begin{aligned} -4 x+4 y & =20 \\ 2 x & =-42 \end{aligned}\right. \]

Solution

Solution Steps

To solve the system of equations using the method of substitution, first solve one of the equations for one variable. In this case, solve the second equation for $ x $. Then substitute this expression for $ x $ into the first equation to find $ y $. If the system is dependent, express the solution set in terms of one of the variables.

Step 1: Solve for $ x $

From the second equation $ 2x = -42 $, we can solve for $ x $: \[ x = \frac{-42}{2} = -21 \]

Step 2: Substitute $ x $ into the First Equation

Now, substitute $ x = -21 $ into the first equation $ -4x + 4y = 20 $: \[ -4(-21) + 4y = 20 \] This simplifies to: \[ 84 + 4y = 20 \]

Step 3: Solve for $ y $

Next, isolate $ y $: \[ 4y = 20 - 84 \] \[ 4y = -64 \] \[ y = \frac{-64}{4} = -16 \]