Solving linear equations with x on both sides
In the previous post, we talked about solving equations by "unwrapping" the layers surrounding x to get it alone. But what about when x shows up multiple times in the equation? What if you have an equation like $7x + 5 = 9(6x - 8) + x - 19$ ?
Our ultimate goal is to end up with $x = number$, but since x shows up on both sides, if we try to "unwrap" the x on the left side, then we end up with a big mess, rather than getting an actual answer.
Before we can work on getting x by itself, we'll have to get all the x's together in the first place. To make that happen, we have two steps we have to do: (1) simplify the left and right sides separately as much as possible, and (2) add or subtract terms to get rid of all of the x terms on one side.
So let's look at how that works with our problem from before, $7x + 5 = 9(6x - 8) + x - 19$. First, we're going to simplify each side separately. The left side is easy: $7x + 5$ is already as simple as it can get, because $7x$ and $5$ are not like terms. The right side is a bit trickier. In $9(6x - 8) + x - 19$, we need to apply the distributive property on $9(6x - 8)$, giving us $54x - 72 + x -19$. Then we can combine like terms, resulting in $55x - 91$. That can't be simplified further, so we're done with this step.
Now that that two individual sides are simplified, we need to add or subtract terms to get rid of x on one side. It doesn't really matter which side we get rid of them on (we should get the same answer either way), so let's eliminate the ones on the right side to end up with x on the left side like the problems we did in the previous post. Since the $55x$ on the right side is positive, we need to subtract $55x$ to get rid of it. And if we're subtracting $55x$ from the right side, then we need to do the same thing on the left side to keep the equation balanced, which means we now have $-48x +5=-91$. And now we can solve that using the "unwrapping" technique of doing reverse PEMDAS - first subtract 5 from both sides to get $-48x =-96$, then divide both sides by 48 to get $x=2$ as our final answer.
Here's one more example of that process in action, using the equation $18(2x - 3)+12 = 5(6x + 10)+4 $. First, simplify each side: on the left side, distribute 18 to get $36x -54+12$, then combine like terms for $36x -42$. On the right, distribute 5 to get $30x +50+4$, and combine like terms to get $30x +54$. Once each side is simplified, add or subtract terms to get all the x's together. Subtracting 30x from both sides to eliminate the x's on the right gives us $6x -42 = 54$. Next, we need to get x alone using reverse PEMDAS. First add 42 to both sides: $6x = 96$, then divide both sides by 6 to get $x = 16$ as the final result.
Now it's your turn! Can you solve these problems? Tell us your answer in the comments!
$4(-14x+1)-19 = 17(-4x-5) + 8$
$2(-16x+11)-17= 3(2x-6) + 4$