How do you solve the system #y=2x+6# and #2x-y=2# using substitution?

Solve the system of equations:

Equation 1: #y=2x+6#

Equation 2: #2x-y=2#

If Equation 2 is converted from standard form to slope-intercept form, you will see that both lines have the same slope, and are therefore parallel lines, and there is no solution.

Equation 2

#2x-y=2#

Subtract #2x# from both sides.

#-y=-2x+2#

Divide both sides by #-1#. This will reverse the signs.

#y=2x-2#

So both lines have the same slope of #2x#.

graph{(y-2x-6)(2x-y-2)=0 [-16.83, 5.67, -19.26, -8.01]}

We are given two equations, one of which has #y# as its subject.

Substitute the expression #2x+6# for #y# in the second equation:

#color(blue)(y = 2x+6)" and " 2x-color(blue)(y) =2#

#color(white)(xxxxxxxxxx)2x -color(blue)((2x+6)) =2#

#color(white)(xxxxxxxxxx)2x -2x-6 =2#

#color(white)(xxxxxxxxxxxxxxxxx)0=8#

This is obviously false and there is no #x# left.
This is an indication that there is no possible solution to this equation.

On closer inspection you should notice that both equations represent equations of straight lines:

#y = 2x +6" and "y = 2x-2#

The lines have the same slope which means they are parallel and therefore will never intersect, hence confirming the answer we obtained before.

First, let's convert our second equation into slope-intercept form

#y=mx+b#

Let's subtract #2x# from both sides to get

#-y=-2x+2#

Next, divide all terms by #-1# to get

#y=2x-2#

We see that our slope, the coefficient on the #x# term, is #2#. This equation has the same slope as the first.

It is important to realize that when two lines have the same slope, it means they are parallel. This means they never intersect.

Since they never intersect, it means this system has no solution.

Hope this helps!