Solving for a Variable on One Side Part 3-Multiplication, addition, and subtraction

[One speaker] 

[Title on the first slide appears on bold yellow]

Female Speaker: Hi, welcome to the video on solving for a variable on one side, but this time, using several operations, specifically multiplication, addition and subtraction.

[Shows a blank screen with a virtual pen tool ready to write]

What i’m talking about is an example like 3x plus 5 equals 17 [writes it on screen]. How do we go about solving this, when we’ve got multiplication going on and addition? What do we do first? Well, the best thing to do is break it down. Recognizing this is saying, “this side of the equation [talks about the side where “3x + 5”] is the same as some number multiplied by 3 plus 5 equals 17” [writes “some number multiplied by 3” and below that writes “Plus 5”, and below that writes “Equals 17”]. Notice what we’ve done is we’ve gone through the order of  operations on this side [refers to the side where “3x + 5”]. We did the multiplication first and then we did the addition, but when solving for a variable, now we work backwards. We work in reverse.

So if the last thing we did was add 5 well then we need to find the additive inverse of five and do that to both sides. This line is basic ally step one of the equation [the line refers to the steps she listed out; multiplication and addition] because we’re following the order of operations. If we follow the order of operations, it says that we have to do the multiplication first and then we do the addition. So, this is basically step one that we would do if we knew what x was and then this [indicating to “plus 5”] is step two [puts “2.” before “Plus 5”] because after doing this multiplication, then we would do addition.

[Removes the third sentence on the screen, which was “Equals 17”]

And these are the steps we would take if we knew what x was in order to find out answer of 17, but when we’re trying to find x, we’re working backwards. So, now we work backwards from step two back to step one. So in this case, we have [starts writing on the screen in blue] the 3x plus 5 equals 17 and now we’ve got to work backwards to isolate x.

These steps [motions the 1st and 2nd step] assume that we knew what x was. Now we’re working backwards to get there. So if we added 5 on this side, we need to add the inverse of 5— the additive inverse—so let’s add a negative 5 to both sides [adds negative to both the sides before and after equals]. Now we have 3x [writes 3x below them all, as she’s solving the rest], well plus 5 minus 5 is zero, so that has cancelled itself out-- equals 17 plus a negative 5 is 12.     

Now we work backwards to the next step where we multiplied by 3. So, since we multiplied by 3, now we multiply by the multiplicative inverse. Sour next step to solve for X is to take ⅓ [writes the next set of numbers below the last one]—because ⅓ is the multiplicative inverse of 3—times 3x equals ⅓ times 12. So ⅓ times 3x is just X because the 3’s cancel. Remember ⅓ times 3 over 1-- 3’s cancel and we’re left with 1. So 1x or just x equals ⅓ times 12 is equal to ⅓ times 12 over 1 which equals 12 divided by 3. 12 divided by 3 is 4. So 4 is our solution for x that will allow this equation to be correct, so 3 times 4 is 12 plus 5 is 17. X equals 4 is the correct solution.

[Screen is blank again]

Let’s do another example. Let’s do [starts writing the numbers again] 5x minus 8 equals 27. Once again, let’s write out the steps that we would take if we already knew what X was. Our first step, using the order of operations would be X, meaning our number, times 5. So some number, multiplied by 5 [writes “x multiplied by 5”]. That's our first step. And then our second step would be subtract 8 [writes “2. Subtract 8”]. So those would be the steps we would use, in order to get our answer of 27. So now lets work backwards. We know that we have 27, we are trying to get back to our variable X. So 5x minus 8 equals 27 [writes it in purple below the second step]. So we need to do the opposite of the second step, which was subtract 8. So we’re adding the inverse of negative 8. So negative 8 plus a positive 8 equals zero [writes the equation on the right side of the screen].

So we know that we need to add a positive 8. So plus 8 and plus 8 to both sides. So now we have 5x a negative 8 plus 8 is zero. We’re left with 5x on the left and 27 plus 8 is 35. So now we go back to the next previous step where we multiplied by 5. So now we need to do the opposite of that. So we multiply by the inverse of 5. [Writes the equation] So ⅕ times x equals 1/5 times 35.  Because we’re doing the same thing to both sides of the equation. And now we solve. So ⅕ times 5 is just 1, [writes the solution on the bottom of the previous equation] so we’re left with 1x or in other words x. Equals ⅕ times 35. Well ⅕ times 35 over 1 equals 35 divided by 5, because 1 times 35 is 35 and 5 times 1 is 5. So 35 divided by 5 is 7. Now let’s see if this solution is the right solution for our equation.

So we bring it back up 5 times 7 is 35 minus 8 is 27. So it is the correct solution. Just to confirm what we've been doing. When we have an equation with multiple operations, we’re solving for a variable, one way to do this is to write out the steps according to the order of operations of how we would solve it if we knew what our variable was. And then we take those steps and we work backwards to find the variable.

[End of video]