Solving Multiple Equations Algebraically at a Glance

Solve this system of equations using substitution:

y = 2x + 5
2y = 4x + 10

In this system the first equation has the y isolated, so we can plug it into the second equation.

2(2x + 5) = 4x + 10

Simplify that thang.

4x + 10 = 4x + 10

Subtract 4x from both sides and things get a little weird.

10 = 10

This is true everywhere, so there are an infinite number of answers.

Solve this system of equation using substitution:

y = 3x + 5
y = x – 3

Since both equations have y isolated, we can substitute either one into the other. Let's substitute the first equation into the second one.

y = 3x + 5
(3x + 5) = x – 3

Now we solve for x:

3x + 5 = x – 3
3x + 5 – x = x – 3 – x
2x + 5 = -3
2x + 5 – 5 = -3 – 5
2x = -8

Dividing by 2, we get x = -4. Now we can substitute -4 into the first equation in place of x.

y = 3(-4) + 5
y = -12 + 5
y = -7

So the solution is (-4, -7).

Solve this system of equation using addition:

y = 3x
-y + 4 = x

This time, we're gonna add the equations: lump all the stuff on the left of the equal signs together, and ditto for the right side of the equal signs.

y + -y + 4 = 3x + x
4 = 4x
1= x

Plug 1 in for x in the first equation, and we get y = 3(1) = 3.

So the solution is (1, 3).