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Solving Systems of Equations by Substitution

Hopefully you have already learned how to solve systems of equations by graphing. If not, view our guide here.

Now we’re going to learn how to solve them by using algebra and a method called substitution.

Again, let’s look at the same two equations:

Equation 1: y=3x-2
Equation 2: y=-4x+5

Notice that both of the equations are isolated for y. If we want to find the point where the two lines intersect, it will be at the identical y-value. Therefore we can simply substitute Equation 1 into Equation 2:

3x – 2 = y = -4x+5
3x-2 = -4x+5

Now you have eliminated the y variable from the system, so you can solve for x.

3x-2=-4x+5
3x+4x-2 = 5
7x = 5+2
7x = 7
x = 1

You’re almost done, but you also have to find the value of y when x=1. Substitute x=1 into either Equation 1 or Equation 2.

y=3x-2
y=3(1)-2
y=3-2
y=1

So the solution is (1,1). This means the two lines intersect at the point (1,1). Graphically, it looks like this: