Solving Equations with Multiplication

Now we're going to work on solving equations with multiplication in them. Keep in mind that multiplication can be represented in different ways. So as we see different types of equations, we want to keep these different symbols in mind.

For example, when we see a number next to something in parentheses, that means that that number is being multiplied by what's in parentheses. Or when we see a number written next to a variable with no symbol in between, that number is the coefficient of the variable, and that means that they're being multiplied together.

In this example, we have seven m equals 63. And you'll see that seven is the coefficient of m. So they're being multiplied together. So we'll need to keep that in mind when we go through the steps of isolating the variable m and figuring out how to undo that seven.

But first, let's make sure that we set up our equation so that we're ready to get to work. I'm going to draw a line down through the equal sign so that I can clearly see the left and right side of the equation.

Next, we want to work on isolating the variable M. To do that, we have to undo the number seven using the opposite operation. And as we mentioned, the seven is being multiplied by M. So the opposite of multiplication is division. So we're going to use division to undo that seven.

Now, notice I use a fraction line to represent division. By doing it this way, we're going to keep our work nice and neat. We're going to show all of our steps written right under each other so that we can clearly see what's going on. And of course, we have to do the same thing on both sides of the equation.

So if I divide by seven on the left, I also have to divide by seven on the right. And now we can simplify both sides of the equation. Now on the left side, we have seven M over seven in a fraction. When we have the same factor in the numerator and denominator, it's going to simplify to just one. So the seven over seven is really equal to one. So that whole fraction on the left side becomes one times M written as one M. Now we work our way to the middle to bring down our equal sign.

Now we can go to simplify the right side of the equation. 63 divided by seven is nine. Now, we're not completely done because our variable m isn't totally isolated. We have one times M. But if you remember back to when we learned about the identity property, when we multiply something by one, it doesn't change its value.

So that one M is really just equal to M. So this simplifies to M equals nine. And now we can tell that our equation is solved because we have those three parts. We have the variable by itself on one side, an equal sign in the middle and a number on the other side.

But before we move on, let's check and make sure that our answer is correct. So first we'll copy down our original equation, seven M equals 63. Then we're going to substitute what we think the answer is, our nine for M into this equation. So we'll have seven times nine equals 63.

And notice that I put the nine in parentheses to indicate multiplication. Now we simplify. Seven times nine is 63. So the left side there will become 63 and we have 63 on the other side. 63 equals 63. Well, that's a true equation. When we get a true equation, that tells us that our answer is correct and that M actually is equal to nine.

For our next example, we have negative eight, b equals 24. Now notice we have that negative sign to the left of the eight there. It's not a subtraction sign. This eight isn't being subtracted away from anything. We treat it as a negative sign. Very important to remember that when we go through our steps of isolating this variable B and undoing that negative eight, we'll go ahead and draw our line down the middle so we can see both sides clearly.

Now we can begin to isolate the variable. As we had mentioned, we have to do the opposite operation. Since negative eight is being multiplied by B, we have to divide by negative eight. Okay, now very important to remember, include that negative sign with the coefficient.

So we have to keep that negative with the eight to show that we are dividing by that coefficient of negative eight. That negative sign isn't going to disappear on its own. We have to be mindful to keep it with that eight. And of course what we do on one side, we have to do to the other side.

So we'll divide by negative eight on the right side as well. And then we simplify both sides. So on the left we have negative eight in the numerator and denominator. So that part is going to simplify to one. So on the left side we'll be left with one times B, brings down our equal sign in the middle.

And then on the right side we simplify 24 divided by negative eight is negative three. And of course when we have one times B, it really just means that we have B there. So we can write this over as B equals negative three and we can tell that it's solved because the variable B is by itself, it's isolated. On one side we have our equal sign and then our number negative three on the other side.

But let's check and make sure that it is correct. Write down our original equation. Now we're going to substitute that negative three for B. We put parentheses around it to show that we are multiplying negative eight times negative three. And now we simplify this to see if we get a true equation.

Negative eight times negative three. Let's see, we're multiplying two negatives. What was the rule that we learned for that? Oh, yeah - when we multiply two negatives that's going to equal a positive. So that left side will become a positive 24. And then of course, we have 24 on the right side. 24 does equal 24. We have a true equation. So our answer is correct. B is equal to negative three.