Solving Linear Equations Using Substitution - Infinitely Many and No Solution

Today we continued learning about substitution and we came across two new concepts: infinitely many and no solution. Just to review, substitution is when you isolate one of the variables and replace the variable in the next equation with that equation. Here is an example:

2x+y=-1
x-2y=12

Now you need to isolate either the x or y in one of the equations by putting it into y=mx+b or x=ky+c.

2x+y=-1
-2x -2x
y=-2x-1

After we have the y isolated we put the equation into the second one which is replacing the y.
Now we have...

x-2y=12 and y= -2x-1
x-2(-2x-1)=12

After that you just distribute and solve.

x-2(-2x-1)=12 Plug x=2 into the other equation to solve for y:
x+4x+2=12
-2 -2 -2x-1=y
5x=10 -2(2)-1=y
5 -4-1=y
x=2 -5=y

The point where the two lines intersect is (2,-5).

It's not all that easy. Sometimes the system has infinitely many solutions and sometimes it has no solution.

Here is an example of a system with infinitely many solutions.

3x+2y=10
-6x-4y=-20

Now you have to take one of the solutions and change it into y=mx+b or x=ky+c to isolate the variable.

3x+2y=10
-3x -3x
2y=-3x+10
2
y=-3/2x+5

Now you can substitute and solve.

-6x-4y=-20
-6x-4(-3/2x+5)
-6x+6x-20=-20
+20 +20
-6x+6x=0
0x=0
0=0

When your final answer is 0=0 that means there is infinitely many solutions. The whole point of solving this system is to figure out where the two lines intersect. If there is infinitely many solutions to the system that means the lines are the same.

A way to prove they are the same is to change both equations to y=mx+b. Here is the system:

3x+2y=10
-6x-4y=-20

We already put the first one into y=mx+b:

y=-3/2x+5

The second one is:
-6x-4y=-20
+6x +6x
-4y=6x-20
-4
y=-3/2x+5

They are the same equations which makes them the same line.

Here is an example of a system with no solution.

3x+y=4
6x+2y=7

Isolate one of the variables, substitute and solve.

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

6x+2(-3x+4)=7
6x-6x+8=7
8=7

When you get two numbers that don't equal each other like 8=7 then there is no solution to the system. You are solving this to find the point where two lines intersect. If there is no solution that means that the lines never intersect which makes them parallel lines.

You can prove this the same way as proving infinitely many solutions: changing both into y=mx+b.

The first one was:
y=-3x+4

The second one is:
6x+2y=7
-6x -6x
2y=-6x+7
2
y=-3x+3 1/2

They have the same slope so they are parallel lines.

Alex S.