Questions: Solve the system of equations by the substitution method. -4x + y = -7 4x - 2y = 4 (5/2, 3) (-2, -9/2) (7/2, 7) (1, 13)

Transcript text: Solve the system of equations by the substitution method. \[ \begin{array}{l} -4 x+y=-7 \\ 4 x-2 y=4 \end{array} \] $\left\{\left(\frac{5}{2}, 3\right)\right\}$ $\left\{\left(-2,-\frac{9}{2}\right)\right\}$ $\left\{\left(\frac{7}{2}, 7\right)\right\}$ $\{(1,13)\}$

Solution

Solution Steps

To solve the system of equations using the substitution method, we first solve one of the equations for one variable in terms of the other. Then, we substitute this expression into the other equation to find the value of one variable. Finally, we substitute back to find the value of the other variable.

Step 1: Solve for $ y $

Starting with the first equation: \[ -4x + y = -7 \] we can isolate $ y $: \[ y = 4x - 7 \]

Step 2: Substitute $ y $ into the second equation

Next, we substitute $ y $ into the second equation: \[ 4x - 2y = 4 \] Substituting $ y $: \[ 4x - 2(4x - 7) = 4 \] This simplifies to: \[ 4x - 8x + 14 = 4 \] which further simplifies to: \[ -4x + 14 = 4 \]

Step 3: Solve for $ x $

Now, we solve for $ x $: \[ -4x = 4 - 14 \] \[ -4x = -10 \] \[ x = \frac{5}{2} \]

Step 4: Find $ y $

Substituting $ x = \frac{5}{2} $ back into the expression for $ y $: \[ y = 4\left(\frac{5}{2}\right) - 7 \] \[ y = 10 - 7 = 3 \]