Video: Solving Linear Equations Mentally Using the Distributive Property of Multiplication

Determine the value of x that makes the equation 7(𝑥 − 3) = 56 − 21 true.

03:55

Video Transcript

Determine the value of 𝑥 that makes the equation seven multiplied by 𝑥 minus three equals 56 minus 21 true.

So, to solve this equation, which is what we will need to do to find the value of 𝑥 that makes the equation true, the first thing we’re gonna deal with is the right-hand side. Because we’re gonna calculate the value of our subtraction. And that is 56 minus 21. To do this, what I’m gonna do is use a written method; I’m gonna use column subtraction. So, we’ve got 56 minus 21.

So, first of all, we’re gonna have six minus one, which is five. And then, we’re gonna have five minus two, which is three. So, therefore, we can say that 56 minus 21 is gonna to be 35. And we could’ve done a quick mental check to make sure it was right. We could do that by adding 21 to 35. Well, if we think about 35, if we add one, we get 36. Then, add 20 gives us 56, which is what we are looking for. So, great, we’ve done that. So, now, what’s the next step?

Well, from this point, there are a couple of methods we could use to solve the equation. So, the first one I’m gonna do is gonna involve the first step being divide both sides of the equation by seven. That’s cause we’re gonna do an inverse operation because we’ve got seven multiplied by 𝑥 minus three. So, the inverse of that is divide by seven, which would just leave us with our 𝑥 minus three. And whatever we do to one side of the equation, we must do to the other.

So, when we do that, we’re left with 𝑥 minus three equals five. And that’s because, as we said, if we divide by seven, we’re just left with 𝑥 minus three on the left-hand side. And 35 divided by seven is five. And that’s because five multiplied by seven is 35. And then, again, what we do is we do an inverse operation. This time, instead of having 𝑥 minus three, what we’re gonna do is add three to each side of the equation. And the reason we’re gonna do that is because if we add three to negative three, then we get zero. And that’s what we want because we want 𝑥 on its own on the left-hand side.

And remember, whatever we do to one side, we must do to the other. So, when we do that, we get 𝑥 is equal to eight. And that’s our final answer because that is the value of 𝑥 that makes the equation true. So, great, we’ve solved it. But what I did say there are other couple of methods that we could’ve used. One of these methods would use something called expanding the bracket. And this is another algebraic skill. And to do that, what we do is multiply the seven by each of the terms inside the bracket.

And when we do that, we get seven 𝑥 for the first term. That’s cause seven multiplied by 𝑥 is seven 𝑥. And then, minus 21, and that’s cause seven multiplied by negative three is negative 21. And this will be equal to 35. Okay, great, so the next stage is going to be add 21 to each side of the equation. And we do that because we’ve got negative 21. And if we add 21 to that, will give us zero. So, we’re left with seven 𝑥 on the left-hand side. And if we do that, we’re gonna get seven 𝑥 is equal to 56.

And we get that because 35 add 21 is 56 cause 35 add one, which we’ve already done earlier, is 36. Add 20 gets us to 56. And then, finally, what we need to do is divide each side of the equation by seven. And if we do that, we get 𝑥 is equal eight, which is great cause that’s the answer we were looking for. And we just solved the problem again. So, that was the other method that I mentioned.

And also a third method that we could’ve used, which would have been a mental method, is we look at seven multiplied by 𝑥 minus three equals 35. And we think, well, seven multiplied by something gives us 35. Well, that must be five. So, that means whatever is in the bracket must be equal to five. So, if we look at that and think, well, what number minus three gives us five? It’s eight. Then, we’d know that 𝑥 is equal to eight. So, that’d be another way of checking using a mental method.

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