How to solve a series of algebraic equations

Sounds scary right? It really isn't, if you follow these simple steps. Let's take the following set of algebraic equations:

3x - y = 1
2x + 3y = 8

So here we have two equations and two unknowns or variables right? There are actually 3 methods for solving a system of equations: graphing, elimination, and substitution. In this blog, I'm going to show you how to solve using the substitution method. Basically, we're going to try to simplify these equations down to one equation with one variable, something we already know how to solve, such as 2x + 1 = 7. The way we're going to do that is listed in the following steps:

Step 1: Solve one equation for either variable in terms of the other variable (it doesn't matter which)
Step 2: Plug this term into the second equation and solve for the other variable
Step 3: Once you have solved for that variable, go back and determine the first variable from the first equation
Step 4: Check your work

So let's take the first equation and solve for y. Remember an algebraic equation is like a balanced seesaw, whatever we do to one side of the equation we have to do to the other side. Therefore subtracting 3x from both sides of the first equation and dividing by -1, we have:

3x - y = 1
-3x - y = 1 - 3x
-y = 1 - 3x
y = 3x - 1

Now, since we know what y is in terms of x, we're going to plug the term (3x - 1) into the second equation and solve for x, so therefore we have:

2x + 3y = 8
2x + 3(3x - 1) = 8
2x + 9x - 3 = 8
11x = 11
x = 1

Now that we have determined x, we can solve for y from our first expression by plugging 1 in for x:

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

Finally, we check our work by plugging in our values for x and y into the first equation and make sure we didn't make any mistakes:

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

Yay, we did it! We just learned how to solve a series of algebraic equations. You can do it!!


Mark M.

Math Teacher with Degree in Chemical Engineering

50+ hours
if (isMyPost) { }