Video: Solving Multistep Equations

By working through a series of examples, we help you move beyond solving one-step equations, such as 𝑥 − 3 = 0, and two-step equations, such as 3𝑥 + 5 = 11, so you can solve multistep equations, such as 9𝑥 − 3𝑥 − 5 = 7.


Video Transcript

Before watching this video, you should be familiar with solving one-step equations like 𝑥 minus three equals zero or five 𝑥 equals thirty and two-step equations like three 𝑥 plus five equals eleven or 𝑥 minus four all over two equals two. We’re gonna move on to consider equations like nine 𝑥 minus three 𝑥 minus five equals seven and 𝑥 plus twelve all over three equals twenty-four over four.

So solving multistep equations is just like solving one-step or two-step equations except there are more steps; there are multiple steps as the name suggests. For example, solve nine 𝑥 minus three 𝑥 minus five equals seven.

Well first of all, we can combine the like terms. So nine 𝑥 minus three 𝑥 gives us six 𝑥. Now we can add five to both sides. And on the left-hand side negative five plus five is nothing, so that just leaves us with six 𝑥. So that was the point of adding five to both sides. And on the right-hand side seven plus five is twelve. Now we just want one 𝑥. So if we divide both sides by six, six divided by six on the left-hand side is one and twelve divided by six on the right-hand side is two. So 𝑥 is equal to two.

Now we can check this answer by substituting the value of 𝑥 equals two back into our original equation. So what we wanna see is does nine times two minus three times two minus five equal seven. Well nine times two is eighteen and three times two is six. So that becomes is eighteen minus six minus five equal to seven. Well eighteen minus six is twelve. So is twelve minus five seven? Well, yes it is. So it looks like 𝑥 equals two was the correct answer.

Now solve 𝑥 plus twelve over three is equal to twenty-four over four. Well the first thing I’m gonna do is put the numerator in parentheses here just to make it clear that the 𝑥 plus twelve belong together on top of that fraction and then I’m gonna try to eliminate the fraction by multiplying both sides by three over one.

And when I do that on the left-hand side, we can divide the top by three and we can divide the bottom by three to just leave us with 𝑥 plus twelve. And while we’re doing some cancelling down on the right-hand side, twenty-four over four, well they’re both divisible by four. Four is going into four once and four goes into twenty-four six times. So the right-hand side becomes six over one times three over one, which is just eighteen.

So that step was really about doing a bit of cancelling down and simplifying. So 𝑥 plus twelve is equal to eighteen. So if I subtract twelve from both sides of my equation, the left-hand side becomes 𝑥 plus twelve minus twelve and the right-hand side is eighteen minus twelve. So on the left-hand side we’ve got plus twelve minus twelve. Well that’s nothing, so those two things are gonna cancel out leaving me with just 𝑥. And on the right-hand side, eighteen minus twelve is six. So 𝑥 equals six is our answer. Now again, we could check this. So what we do is substitute the value 𝑥 equals six back into our original equation. We want to check that it’s six plus twelve all over three that equals twenty-four over four. Well six plus twelve is eighteen.

So we’re trying to find out if eighteen over three is equal to twenty-four over four. Well we can do a bit of cancelling. On the left-hand side, eighteen and three are both divisible by three: three divided by three is one; eighteen divided by three is six. And on the right-hand side, four is divisible by four. Fours in four go once and twenty-four is divisible by four as we said before and that goes six times. So we have- we’ve got six over one equals six over one. So yes it does, and we’ve got the right answer.

Well this looks like a pretty straightforward question. But the fact that it’s minus three lots of 𝑥 gives us an extra step that turns it into a multistep question. So solve thirty-five minus three 𝑥 is equal to fourteen.

But now we’ve seen a few examples already. I’m gonna start putting in a little bit less of my working out down so that we see a slightly more compact form of our working out. So thirty-five minus three 𝑥 is equal to fourteen. I’m going to add three 𝑥 to both sides of my equation to get rid of that negative number of 𝑥s. Now when I add three 𝑥 to the left-hand side, I’ve got thirty-five minus three 𝑥 plus three 𝑥. They’re gonna cancel out, so that’s just gonna leave me with thirty-five. When I add thirty- when I add three 𝑥 to the right-hand side, I’ve got fourteen plus three 𝑥. Now I can subtract fourteen from both sides.

And when I subtract fourteen from thirty-five, that leaves me with twenty-one. And if I subtract fourteen from the right-hand side, I’m just left with three 𝑥. Well that’s what three 𝑥 is. I want to know what one 𝑥 is. So I want to know what a third of that is. So dividing both sides by three, a third of twenty-one is seven and a third of three 𝑥 is just one 𝑥. So that’s our answer. And again, I can check my answer if I’ve got time by substituting that value 𝑥 equals seven back into our original equation and seeing if it actually satisfies the equation.

Now our last question is given that 𝑥 plus two 𝑥 plus three 𝑥 plus four 𝑥 plus five 𝑥 equals a hundred and twenty, find the value of four 𝑥 minus two. So in this problem, we’ve got a multistep problem to actually find the value of 𝑥, and then we’ve got to substitute that value of 𝑥 into this expression to find the value of four 𝑥 minus two. So we’ve gotta be careful that we actually answer the question and we don’t just find the value of 𝑥.

So to start off with, we’ve got 𝑥 plus two 𝑥 plus three 𝑥 plus four 𝑥 plus five 𝑥 equals a hundred and twenty, and we can combine all those like terms. So we’ve got one 𝑥, then another two 𝑥s, then another three, then another four, then another five. That makes a total of fifteen 𝑥s and that’s equal to a hundred and twenty. Now I want to know what one 𝑥 is. So I’m gonna divide both sides by fifteen. And a fifteenth of fifteen 𝑥 is just one 𝑥 and a hundred and twenty divided by fifteen is eight.

But as we said before, that’s not actually answering the question. So 𝑥 is equal to eight. We’ve now got to substitute 𝑥 equals eight into the expression four 𝑥 minus two to evaluate that. So when 𝑥 equals eight, then four 𝑥 minus two is four times eight minus two, and four eights are thirty-two. And thirty-two minus two is thirty. So our answer is four 𝑥 minus two equals thirty.

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