Learn to Solve Linear Equations in 10 Minutes

Getting Started Without Losing Your Mind

One of the most basic and most useful skills that you learn in Algebra 1 is how to solve linear equations. There is a general strategy involved in this process, and if you keep this strategy in mind, you'll be able to solve really hard equations without breaking a sweat. Before we get started actually solving equations, we're going to look at the one rule of Algebra 1 that you'll need to know to be able to put this strategy to work.

Here's the rule: if you do an operation to one side of an equation, you have to do the same operation to the other side of the equation for it to remain true.

So here's an example of what we mean by this. If we look at a simple equation like 3 + 2 = 5, then we can subtract 1 from the left side of the equation if we really want to and we'll have 3 + 2 - 1 on the left side. Because we subtracted 1 from the left side of the equation, we'll also have to subtract 1 from the right side of the equation to keep it balanced. This means we'll have 5 - 1 on the right side of the equation, and our new equation will be 3 + 2 - 1 = 5 - 1. If you simplify both sides, you'll see that you get 4 = 4, which is certainly true.

One more example. Let's say we have the equation 4 + 5 = 9, and we want to multiply both sides by 2. On the left side we'll have 2(4+5), since we're multiplying the entire side by 2, and on the right side we'll have 2(9). This gives us the equation 2(4+5) = 2(9), which simplifies to 18 = 18.

Once you know this rule, then it's time to learn the one Algebra 1 strategy that will help you solve all linear equations. Take the short quiz below to make sure you understand these ideas before moving on.

Quiz for Solving Linear Equations

Get the Variable By Itself on One Side of the Equation

Once you understand the rule of algebra having to do with balanced equations, you can put the rule to work by using the following strategy:

Get the Variable By Itself on One Side of the Equation

You're going to do this by eliminating everything on one side of the equation that isn't the variable. Let's look at an example. Suppose we want to solve the equation

2x - 5 = 1

We would like to get an answer that looks like x = ??, since that will just tell us what x is in a very straight-forward manner. We need to get rid of everything on the left side of the equation that isn't x. Let's start with the subtraction of 5. The opposite of subtraction is addition, so if we add 5 to the left side of the equation, it will cancel out. However, our rule about balanced equations tells us that we'll have to add 5 to the right side of the equation as well. No problem! Here's what we get:

2x - 5 = 1
2x - 5 + 5 = 1 + 5
2x = 6

Now that the subtraction of 5 is out of the way, we have to get rid of the multiplication by 2. The opposite of multiplication is division, so if we divide the left side by 2 the multiplication by 2 will cancel out. However, if we divide by 2 on the left side of the equation, we have to divide by 2 on the right side of the equation. This isn't a problem, it's just what we have to do to keep the equation balanced. So here's what we get:

2x = 6
2x/2 = 6/2
x = 3

And we see that x = 3.

The Grand Strategy

An Example Problem Worked Out

Equation
Steps
4x - 11 = 5
Starting Equation
4x - 11 + 11 = 5 + 11
Add 11 to both sides
4x = 16
Simplify
4x/4 = 16/4
Divide by 4 on both sides
x = 4
Simplify. We're done!

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