Solving equations is a crucial skill, especially for high school math. On a certain level, solving equations is the easiest way to describe all of high school math. The goal for this post is to go over simple single variable equations, complex single variable equations, simple system of linear equations, and complex system of linear equations. If you would like to learn more about parabolas, please refer to this POST where I take a deep dive into quadratic equations and more importantly the information you can get out of them. Regardless, let’s move forward with our review of solving equations!

## Simple Single Variable Equations

To begin, we have simple single variable equations. These are very easy to work with, but unfortunately stop occurring after grade 9. Anyways, these kinds of equations have the form ax + b = c. Some examples include:

When it comes to solving equations, the biggest mistake students make is not knowing when to make use of multiplication/division and when to use addition/subtraction. The most important thing to remember is addition/subtraction are used in the goal of isolating a variable on one side and getting constants on the other. Moreover, multiplication/division are more commonly used in getting rid of fractions and dividing by the coefficient of a variable to solve for its value. With simple single variable equations, it will usually involve 1-2 steps. The first being addition or subtraction to get ax = c and the second being dividing by ‘a’ to get the value of x (or whatever variable is being used). Let’s look at solving those equations above using this logic:

As you can tell, these kinds of simple equations are very basic and require minimal effort to get to the answer. However, with complex single variable equations, the work goes up a bit.

## Complex Single Variable Equations

With these equations, the biggest difference is the additional use of subtraction/addition to isolate for variables along with multiplication to get rid of fractions. Let’s take a look at what these problems could look like:

This is probably what some of the harder single variable equations look like. To be honest, once you get the hang of brining the equation to the form of ax = c, it’s not too difficult. Without further ado, let’s go through these problems. Well, some new steps required would be getting rid of fractions and adding/subtracting variables to isolate them on one side. Now, let’s go through the solutions:

As a result of the increased complexity, the process is a bit longer, but the fundamental approach is the same. Furthermore, we have to make proper use of addition, subtraction, multiplication, and division to get to: ax = c. For example, with the first problem, if we divided by 10 to get x one side, it wouldn’t help us. The reason being, we’d get this: 0.7x + 1.2 = x. The problem is that we have the exact same format and we haven’t gotten closer to the required format. Now, let’s go on to a simple system of linear equations!

## Simple Systems of Linear Equations

When it comes to a system of linear equations, there are 3 general approaches. 1) Elimination, 2) Substitution, and 3) Graphing. I prefer elimination, but some people do have contrary views. For this post, we’ll stick with elimination and substitution. However, we will use graphing to check our answers. Now, here are some base problems we’ll go through:

To be clear, elimination works on the idea that you either add or subtract equations to get to a point where you have ax = c. For example, with question 1, if we add those 2 equations together, we’ll get an equation of the form ax = c. From there, we can solve for x, plug that value back into an equation and solve for y. Similarly, with problem 2, we can subtract the second equation from the first and get the value of x.

For fun, we can put our equations in an online graphing tool, such as desmos.com, in order to check our POI values:

Overall, simple systems are very easy to work with. It is just like a linear system with the exception of plugging back one variable to find the other. Unfortunately, getting easy questions like this is rare, so let’s work on our complex ones below:

## Complex Systems of Linear Equations

Like all of our examples, let’s begin with some problems to work on:

When our equations don’t give us easy ways to eliminate or solve for one variable, we can do one of two things. Primarily, we can multiply both equations by a certain value such that the coefficients of one variable are the same in both. For instance, with problem 1, we can multiply equation 1 by 4 and equation 2 by 3 to get a common value of 12y in both. This will allow us to subtract and then solve for y. The second option is substitution. We can simply find what one variable is equivalent to in one equation and then plug that into the other equation to get a simple/complex single variable equation. Hence, let’s apply these concepts to our problems!

A little more complicated and quite lengthy, but the fundamentals haven’t changed. It’s just a couple extra steps to simplify the equations. Finally, let’s check our answers with desmos:

Wow! Our answers matched both of the graphs. To conclude, it is very important to understand how to solve these kinds of equations, especially for high school. It can be difficult at times, but remember what format you’re trying to get to (ax = c). Also, before you go, be sure to check out my post on how I created a different way to solve a system of linear equations: LINK, my post on slope and the different forms of linear equations: LINK, and finally my post on parabolas and quadratic equations: LINK!