Solving a Variable on One Side With Many Operations

[The following equation appears on the screen. Narrator writes work as she speaks is.]

Woman Narrator: Welcome in this video I am going to go over another example of solving for X on one side of the equation. This time with a lot of different operations going on. But we’re going to break it down step by step and you’ll see by doing this you too can solve any equation no matter what operations are going on. So, here is our variable X and we’ve got to get it all by itself on one side of the equal sign. But it’s surround by all of this other stuff. So, the first thing we need to go is figure out how we got from X to the answer 20. And we do that as though we know what X is and follow the order of operations. So, what would be the order of operation to do a problem that looks like this? So, the first thing we need to do is look at this equation. Well we have parentheses here on this center section so, we know right away we need to do that first whatever is in the parentheses first. Now within the parentheses we have a lot going on. In fact we have a fraction right here and that implies that there's division going on. Division is very important it comes before addition or subtraction.

So, let's look at the fraction first [Referring to the fraction, (-2x+8)/5]well within the fraction we actually have another implied set of parentheses right here. Even though there isn’t a set of parentheses around it it’s as though there is. Because all of this on the numerator is being divided by five. So, everything up here is being grouped together so we can then divide by five. So it’s as though there are parentheses around this portion as well in the numerator. So, the numerator of this fraction within the entire equation is where we need to start. And then even within the numerator we have multiplication and addition going on. So, the very first step we have assuming we know X and we’re following the order of operations. Our first step is X times or multiplied by  negative two.

Our next step would be to add eight. Now we’ve put everything in the numerator together. So now we can divide the entire numerator by five. So, we’ve done all of this the fraction part and now we can go onto subtracting three. So, now we’ve done everything within the parentheses. So, now we can look outside the parentheses and we have three is being multiplied to everything within the parentheses. And then we’re adding seventeen so, multiplication comes before addition. So, our fifth step is to multiply by three. And our sixth step is to add seventeen. So, assuming we know what X is if we first multiply X by a negative two and then add eight and then divide all of that by five and then subtract three and then multiply by three and then add seventeen we should get twenty.

 But since we don’t know what X is now we’re gonna work in reverse. We’re gonna undo all of these steps. Starting from step six and working back to step one in order to solve for X. So we’re gonna do the opposite of each of these steps to both sides of the equation. So now I’ve shrunk it a little bit to give us a little bit more room. So, let's start at the bottom of our list and work backwards to solve for X. So, up here we added seventeen so we need to add the inverse of seventeen. So we add a negative seventeen on the left of the equal sign and a negative seventeen on the right side of the equal sign. So, now let's rewrite our equation. So, on this left hand side we still have three times in the parentheses negative two X plus eight divide by five minus three all in parentheses. Now our positive seventeen plus a negative seventeen equals zero and anything plus zero is just itself so we can leave that off.

[Writes the equation,

So, that is equal to the equal sign well twenty plus a negative seventeen is three. Because twenty minus seventeen is three. So, step six is done we’ll check that one off. So, now lets work backwards and do the opposite of step five. So originally the step was to multiply by three. So, now we want to multiply by the inverse of three. So, we’re gonna multiply both sides of this equation and we’re gonna use brackets to show I’m multiplying. We’re gonna multiply by one third because that is the inverse of three. And do the same thing to the right hand side one third. Well we can ignore the stuff in the brackets and just say three times one third is equal to one. So, this three and this three cancel and actually on this other side we have three times one third is actually a one as well. Because a three divided by three is one. So, let’s rewrite our equation again.

So, now we have one times everything in parentheses so we can just write everything in the parentheses by itself because one times anything is just itself. So, negative two X plus eight divide by five minus three is equal to this side three divide by three is equal to one.[Writes the equation, ] So, there we go we’ve checked off step number five. So, now we go back to step number four. Step number four says subtract three so we see this number right here. So we need to add the inverse of a negative three so the inverse of negative three is positive three. So if we had three to both sides add three to both sides. We’re doing the opposite of step number four and let’s rewrite our equation. So, we’re left with a negative two X plus eight divided by five minus three plus three is zero. So we don’t need to write that equals one plus three is four. So we have done step number four now or the opposite of step number four.

So, now we need to do the opposite of step number three divide by five. So, right here we’re dividing by five so if we do the opposite of that we want to multiply by five or in other words right here this divide by five is actually the same as multiplying by one fifth. So, if we multiply both sides by five that is the opposite of dividing by five. Multiply both sides by five and so our fives cancel on the left hand side we’re left with four times five is twenty. So let’s rewrite this one more time we have on the left hand side we have negative two X plus eight equals twenty. Each time we finish one step we rewrite our equation. And we’re done now with the opposite of step number three.

So, now we’re up to step number two add eight. So, we need to do the opposite of adding eight. So the opposite of adding eight is to subtract eight or in another words add a negative eight. So, if we add a negative eight to the left hand side we also add an negative eight to the right hand side. So positive eight plus negative eight is zero and we can rewrite this equation again. So, now on the left hand side we have negative two X plus eight minus eight is zero equal to twenty plus negative eight is twelve. Twenty minus eight is twelve. And the opposite of step number two is done.

So finally we need to do the opposite of step number one which was to multiply by negative two. So if this step was to multiply by negative two the opposite or inverse is to multiple by negative one half and the negative stays with it. Multiply by a negative one half to the right side half as well. Because the inverse of negative two is negative one half.

So, here’s a quick explanation of why the inverse of negative two is negative one half. Remember negative two is the same as negative two divided by one. But this negative could actually be applied to either the numerator or the denominator or we could even put it out in front it all means the same thing. But to find the inverse we take the numerator and we put it in the denominator so a negative two into the denominator and the denominator into the numerator. So, the inverse is one over a negative two but remember what I said the negative could either be in the denominator or in the numerator or even out front. One over negative two is equal to negative one divided by two.

So, we have one other thing going on here where a negative times a negative becomes a positive. So, not only can we cross out our twos but our negatives are also gonna go away. Because a negative times a negative becomes a positive. So we are left with X all by itself on the left hand side of the equation which is equal to twelve times a negative one half is the same as twelve divided by a negative two equals twelve divided by negative two equals negative six.

So, X equals negative six is the solution to this equation that we started with way up here that may have looked a little bit daunting at first but by breaking it down step by step and then undoing each of those steps by going backwards we were able to solve for X using the skills we had learned. Now it took some time we took a lot of steps we had to do a lot of different calculations but by taking it slowly and just going step by step by step we were able to find the solution.

[End of video.]