# Video: Solving Linear Equations with Unknowns on Both Sides

We solve a series of linear equations with the unknown on both sides, such as 3𝑥 − 5 = 2𝑥 + 3 or 1.5𝑥 + 5 = 3𝑥 − 1. We look at strategies for rearranging the equation in order to make the working as simple as possible.

15:48

### Video Transcript

In this video, we’re gonna see how to solve linear equations in which there are terms involving the unknown variable on both sides of the equation. With a bit of careful thought, we can make things easier for ourselves by ensuring we end up with a positive number of the unknown on one side of the equation. Let’s take a look at some examples and find out more.

Number one, solve three 𝑥 minus five equals two 𝑥 plus three. Well as you can see, we’ve got three 𝑥 on the left-hand side and we’ve got two 𝑥 on the right-hand side. So our first task is to try to eliminate 𝑥 from one of the sides. We’ve got the choice of two inverse operations. We can either subtract three 𝑥 from both sides to eliminate 𝑥 from the left-hand side or we can subtract two 𝑥 from both sides to eliminate 𝑥 from the right-hand side. Now if I subtract three 𝑥 from both sides, on the left-hand side I’ve got three 𝑥 take away five take away three 𝑥, and three 𝑥 take away three 𝑥 is nothing. So that’s just gonna leave me with negative five. Then on the right-hand side, I’ve got two 𝑥, and I’m taking away three 𝑥. And two 𝑥 take away three 𝑥 is negative one 𝑥, or just negative 𝑥. And then obviously, I’ve got plus three as well. Well now I’ve got negative five is equal to negative 𝑥 plus three. I’ve got a choice of two things. I can either try to clear off the three from the side that’s got the 𝑥 on it, but then that will still leave me with a negative number of 𝑥s. Or I can add 𝑥 to both sides to get a positive number 𝑥s on one side. And I’m gonna try adding 𝑥 to both sides. So on the right-hand side, I’ve got negative 𝑥 plus 𝑥, which is nothing. So those two to cancel out, just leaving me with three. Now I’ve got negative five plus 𝑥 on the left-hand side is equal to three. So if I add five to both sides, then on the left-hand side I’ve got negative five plus five, which is nothing. So they cancel out, just leaving me with 𝑥. And on the right-hand side, three plus five is eight, so I’ve got 𝑥 equals eight.

Okay, so that works. I’ve got my answer. Let’s rerun that again. But this time instead of taking away three 𝑥 from both sides in the first stage, let’s take away two 𝑥 from both sides. So on the left-hand side, I’ve got three 𝑥 take away two 𝑥 is just 𝑥. And obviously, I’ve still got the minus five term. And on the right-hand side, I’ve got two 𝑥 take away two 𝑥. Well that’s nothing, so they cancel out. So that just leaves me with positive three. Now I can just add five to both sides to get rid of the negative five from the left-hand side. And that means on the left, I’ve got negative five add five, which is nothing. So they cancel each other out, just leaving me with 𝑥. And on the right-hand side, three plus five is equal to eight. So we came up with the same answer in both cases. So I’ve come up with the same answer, 𝑥 equals eight in both questions. But when I eliminated the two 𝑥 from this side and left myself with a positive number of 𝑥s on the other side rather than the negative number of 𝑥s, the whole thing was quicker. I eliminated the need for one of the steps of my calculation. So there’s a top tip; when you’ve got a question with the unknown on both sides of the equation, try to eliminate the one that’s gonna leave you with a positive number of that variable on the other side of the equation, and it would just save you a bit of time.

Okay, let’s move on to another question.