Solving Multi-Step Equations
High School / Math / Pre-Calculus
Today we're going to take our equations unit a little bit further and start solving multi step equations. Multi meaning more than one. Now we've already done one step equations and we've already done two step equations. Now we're just adding a little bit to it. So when we're solving multi step equations, we follow those same rules of isolating the variable. That's our goal to get our variable all by itself. We think of it as undoing operations. Our operations meaning addition subtraction multiplication and division. There are 5 steps to follow when solving a multi step equation. Step one, distribute if possible. So if we have any distribution, we have to do that first. Don't forget your negatives. Step two. Simplified each side of if possible, meaning we combine like terms on each side of the equation. Step three isolate the term with the variable. Now this should look very familiar because we did this for two step equations. Isolate that term with a variable. Again, with two step equations, our goal was to isolate the variable, which I got a final goal here. And then I did say 5 steps because you should always check your answer, make sure it's reasonable. So just take a look at your answer and be like, does that really make sense? Does it work? Okay? So let's do a couple of examples. I'm going to keep our steps there so we can follow it. Example one, three X plus two times in parentheses, 5 X plus 16. Equals 162. So step one says distribute if possible. Well, we can definitely distribute here. Since we have a positive two, because it's plus two. I don't have to worry about keep changing change or worry about negatives for right now. So I'm going to distribute. I end up with three X, which just comes down. Plus two times 5 X is ten X plus two times 16 is 32. That's still equals 162. I'm just going to draw by line here, so I can make sure to stay balanced. Okay. So we've done step one. There's no more distributive property to follow. We got rid of the distributive property. Step two, simplify each side if possible by combining like terms. Well, on the left hand side here, I have some like terms. I have three X plus ten X. I can combine those. Three X plus ten more X is 13 X's. Plus 32 can't be combined with anything on this side. Make sure you stay when you simplify. You stay on one side of the equation at a time. So I'm just going to bring down my 32. Equals 162s all alone. I can't combine that with anything. So I'm just going to bring it down. All right. Now we've simplified each side. I only had to simplify the left hand side here, but we've definitely done both sides. Next step, isolate the term with the variable. 13 X is my charm with a variable. I want to get that all by itself. That's why I have to get rid of this plus 32. I'm going to subtract 32 on both sides. To isolate the charm with the variable. Now remember, if you do something on one side, you have to do it to the other. We can combine all we want. But once we start an operation, adding subtracting and multiplying or dividing on one side, you got to do it to the other. So my 32s become zero. I'm left with 13 X equals 162, take away 32. It's 130. Finally, I slit variable. Divide both sides by 13. To get that X all by itself, because remember 13 over 13 is one whole. So we get X equals 130 divided by 13 is ten. X equals ten. And then we'd say to ourselves, does that make sense? Is that reasonable? Well, let's go back to our original equation. What if I use substitution and put ten in for each of these? Well, three times ten will be 30. And then on the inside of the parentheses, 5 times ten would be 50 plus 16 more is 66. Times two is in the 121 130 range. 30 plus about a 120 or a 130 is just about a 162. So our answer is definitely reasonable. So that makes sense. Now what happens when you
When we?re solving multi-step equations, we follow the same rules of isolating the variable. That?s our goal. To get our variable all by itself. We think of it as undoing operations. Our operations meaning addition, subtraction, multiplication and division.
There are five steps to follow when solving a multi-step equation.
Step one. Distribute if possible. So if we have any distribution we have to do that first. Don?t forget your negatives.
Step two. Simplify each side if possible. Meaning we combine like terms on each side of the equation.
Step three. Isolate the term with the variable. Now this should look very familiar because we did this for two-step equations. Isolate the term with the variable.
Again with two-step equations our goal is to isolate the variable. Which I verified all the way here. And then I did say five steps because you should always check your answer. Make sure it?s reasonable. So just take a look at your answer and be like does that really make sense. Does it work?
Okay so let?s do a couple of examples. I?m gonna keep our steps there so we can follow it. Example one, three X plus two, time in parentheses five X plus sixteen equals one hundred sixty-two. So step one says distribute if possible. Well we can definitely distribute here. Since we have a positive two because it is plus two, I don?t have to worry about keep change change or worry about negatives right now. So I?m gonna distribute. I end up with three X which just comes down, plus two times five X is ten X plus two times sixteen is thirty-two that still equals one hundred sixty-two. I?m just gonna draw my line here so I can make sure I stay balanced. Okay.
So we?ve done step one. There is no more distributive property to follow. We got rid of the distributive property. Step two is simplify each side if possible by combining like terms. Well on the left hand side here I have some like terms. I have three X plus ten X. I can combine those. Three X plus ten more Xs is thirteen Xs plus thirty-two can?t be combined with anything on this side. Make sure when you?re simplifying, you stay on one side of the equation at a time. So I?m just gonna bring down my thirty-two. Equals one hundred sixty-two is all alone, I can?t combine that with anything so I?m just gonna bring it down. Alright.
Now we simplify each side. I only have to simplify the left hand side here but we?ve definitely done both sides.
Next step, isolate the term with the variable. Thirteen X is my term with the variable I wanna get that all by itself. So I have to get rid of this plus thirty-two. I?m gonna subtract thirty-two on both sides to isolate the term with the variable. Now remember, if you do something on one side you have to do it to the other. We can combine all we want but once we start an operation - adding, subtracting, multiplying or dividing - on one side, you got to do it to the other. So my thirty-two becomes zero. I?m left with thirteen X equals one hundred and sixty-two take away thirty-two, it?s one hundred thirty.
Finally, isolate the variable. Divide both sides by thirteen to get that X all by itself because remember thirteen over thirteen is one whole so we get X equals one hundred thirty divided by thirteen is ten. X equals ten.
And then we say to ourselves does that make sense, is that reasonable? Well let?s go back to our original equation. What if I use substitution and put ten in for each of these? Well three times ten would be thirty.Then on the inside of the parentheses, five times ten would be fifty plus sixteen more is sixty-six, times two is in the one hundred twenties one hundred thirties range. Thirty plus a hundred twenty or a hundred thirty is just about a hundred and sixty-two. So our answer is definitely reasonable. So that makes sense.
