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Video: Solving One-Step Equations

Tim Burnham

We take a detailed look at how to solve a series of one-step linear equations, such as x - 3 = 10. We think about how to clearly present our mathematical thinking through what we write down.

10:10

Video Transcript

In this video, we’re gonna be solving a series of one-step linear equations, like the one shown in the example here. Okay, let’s dive in and do our first example. Solve 𝑥 minus three equals ten. So when you see questions that say solve something, in this case they mean find the value of 𝑥 that satisfies that equation. So what number is it that when I subtract three from it I get an answer of ten?

Now when you see an equal sign you need to think of this as being a balance; what’s on the left-hand side of the equal sign is balancing what’s on the right-hand side. So if they were on a pair of scales, then the scales would balance perfectly. And what we’ve gotta do is use reciprocal operations in order to get that 𝑥 on its own on the left-hand side of the balance. And the reciprocal operation for subtracting is adding, and the reciprocal operation for adding is subtracting. And, likewise, multiplying and dividing are reciprocal operations; to undo one, you do the other.

So how am I gonna get rid of this negative three from the left-hand side? Well it means take away three, so the reciprocal operation for taking away three is adding three. So I’m gonna add three to both sides of my balance. So I’ve added three to the left-hand side and three to the right-hand side; those two things still balance. But now on the left-hand side, we’ve used reciprocal operations, so I’ve got 𝑥 take away three then add three; if I take away three then add three, those two things cancel out, so I’m just left with 𝑥. So on the left-hand side we’ve just got 𝑥 and ten plus three on the right-hand side is thirteen. So the answer is 𝑥 is equal to thirteen and that makes sense; thirteen take away three is equal to ten.

Now luckily you don’t have to do the drawings and add the balance of the scales every time you work out these one-step equations, but it definitely helps to write some working out as you go through the question. So let’s look at number two: if 𝑥 plus two equals five, then what is the value of 𝑥? So this time, we’re not gonna draw the scales or the balance, but we are gonna write down the operations that we’re doing each time. So on the left-hand side, we’ve got 𝑥 plus two. Well the reciprocal operation of adding is subtracting; so in order to get rid of that plus two, I’m going to subtract two. And remember, I need to subtract two from both sides of the equation to keep it balanced. So on the left-hand side, we’ve got 𝑥 plus two, but we’ve now subtracted two from that; and on the right-hand side, we’ve got the five, but we’re subtracting two from that as well.

Now two take away two is nothing, so on the left-hand side we’ve just got 𝑥. And five take away two is three, so on the right-hand side we’ve got three. So 𝑥 is three. So replacing 𝑥 in our original question with three, we’ve now got three plus two equals five; and yep that’s right. So it looks like we got the right answer.

Number three: solve 𝑥 plus seven equals seven. So the reciprocal operation for adding is subtracting. So in order to get rid of that added seven on the left-hand side, I’m gonna subtract seven from both sides of my equation. And when I do that, the left-hand side becomes 𝑥 plus seven minus seven. So they’re gonna cancel out, which just leaves me with 𝑥 on the left-hand side. And on the right-hand side, seven take away seven. Well they cancel out as well to make zero. So in this case, our answer was 𝑥 equals zero and that’s something that catches out lots of people. It’s perfectly okay for the answer to be 𝑥 equals zero. So putting 𝑥 equals zero back into our original equation, zero plus seven is equal to seven. Yeah that’s right; that’s the correct answer.

Number four: if 𝑥 plus eight equals five, find the value of 𝑥. So we’re going to use reciprocal operations. And the reciprocal of adding eight is subtracting eight, so I’m gonna subtract eight from both sides of my equation. So on the left-hand side, I’ve got my 𝑥 plus eight and now I’m subtracting eight; now eight take away eight is nothing, so I’m just left with 𝑥. And on the right-hand side, five take away eight gives me negative three. If I start off on five and I take away eight — one, two, three, four, five, six, seven, eight — I end up at negative three. So it’s okay for 𝑥 to be a negative number or a positive number or zero. Let’s just do a check. Putting 𝑥 equals negative three back into original equation negative three plus eight equals ? Well we start off on negative three and we add eight — one, two, three, four, five, six, seven, eight — we end up at five. That’s good; that’s the answer we were looking for. So 𝑥 equals negative three is correct.

Number five: solve 𝑥 over three equals five. Now remember, 𝑥 over three means 𝑥 divided by three, so we’re looking for the reciprocal operation of dividing by three and that is multiplying by three. So I’m going to multiply both sides of my equation by three. Now on the right-hand side, five times three is fifteen; and on the left-hand side, I’ve got 𝑥 divided by three times three and that’s just 𝑥.

Another way to think about this is another version of three is three over one; three divided by one is three, so three over one is the same as three. It’s just the fraction version if you like. And when I’ve got a fraction times a fraction, I can cancel what’s on the bottom with what’s on the top if they’re divisible by the same thing. So three divides by three to make one and three divides by three to make one. So I’ve got 𝑥 times one and one times one so that is just 𝑥. So our answer is 𝑥 equals fifteen.

Now if we substitute that number back into the original equation to do a check: we said 𝑥 is fifteen so that becomes fifteen divided by three; and yes that equals five. So that is the number we were looking for, so the answer is 𝑥 equals fifteen.

Number six: solve four 𝑥 equals twelve. Now remember four 𝑥 a four right up against the 𝑥 means it’s four times 𝑥. So we’re looking for the reciprocal operation of four times, and that is dividing by four. So on the left I’ve got four times 𝑥 divided by four and I’ve got a four on the top and a four on the bottom. They both divide by four, so I’ve got one times 𝑥 over one, which is just 𝑥; and on the right-hand side, I’ve got twelve divided by four which is just three. So my answer is 𝑥 equals three.

And now let’s just check our answer by substituting it back into the original equation four times — well we said 𝑥 is three — so four times three. And that is equal to twelve. So that’s what we were looking for; that’s looks like the correct answer.

And lastly, number seven: solve three 𝑥 is equal to negative fifteen. And remember three 𝑥 means three times 𝑥, so the reciprocal operation is gonna be not times-ing by three but dividing by three. So I’m gonna divide the left-hand side by three, and I’m gonna divide the right-hand side by three. So on the left-hand side, three divided by three is one, three divided by three is one, so I just got one 𝑥 over one, which is just 𝑥. And on the right-hand side, I’ve got negative of fifteen divided by three so that’s negative five. So our answer is 𝑥 is equal to negative five. And when we check that we have three times negative five or three times five would be fifteen. So a positive times a negative makes a negative, so the answer is negative fifteen. And that is exactly what we were looking for, so it looks like that’s correct.

Now I know we said this was a one-step equation, but there are three steps to solving one-step equations. Step one is just to identify the reciprocal operation; so for example in an equation like 𝑥 plus two equals five, take away two is the reciprocal of adding two, and so step two is just to do that to both sides of the equation. So here we would subtract two from the left-hand side and we would subtract two from the right-hand side, which would give us 𝑥 is equal to three. And the third step in our one-step equation is to check your answer. So the equation was 𝑥 plus two equals five, and we think that 𝑥 is equal to three. So we can just put three plus two equals five, and it does in fact equal five, so we think we’re correct.