Solving Systems of Equations by Using Substitution

Systems of equations are used to solve a set of equations consisting of two unknown variables. In this post, I will explain how to solve such a set of equations using the substitution method.

Example:

x - y = 0

5x - 3y = 10

First, we need to isolate one of the variables in either of the equations. I will substitute x in the first equation by adding y to both sides.

x = y

Next, we can substitute the value of x in terms of y into the second equation. Since x = y, the second equation can be expressed as the following:

5y - 3y = 10

2y = 10

y = 5

Now that we know the value of y, we can substitute this value into either of our two original equations in order to get the value of x. I will use the first equation.

x - y = 0

x - (5) = 0

x = 5

At this point, we have determined that x = 5 and y = 5. We can substitute these values into either of the two original equations in order to determine if we have gotten the correct result.

x - y = 0

(5) - (5) = 0

0 = 0

Through this process, we have concluded that both x and y are equal to 5.

Also, systems of equations can have three different outcomes:

1. One unique solution: the graphs of the equations intersect at one point.

2. No solution: the graphs of the equations never intersect.

3. Infinitely many solutions: the graphs of the equations are the same.