Solving Equations with a Variable on Both Sides - Problem 3
In any single variable equation, start by simplifying -- distribute and combine any like terms on both sides of the equation. Now you can start solving using inverse operations. The goal is to get the variable on one side of the equal sign and all the numbers on the other side. Since there are variables on both sides, you want to eliminate the variable from one side of the equation. After you have done this, the variable should only be on one side of the equal sign. Next, you want all the constants on the other side of the equal sign. Use inverse operations to eliminate any constants that are on the same side of the equal sign as the variable. Remember that when solving equations, you must work in the reverse order of PEMDAS. In some problems after you bring the variable to one side and all the constants to the other side, the result will be an equation that is not true. In other word, the result will be an equation in which one side does NOT equal the other side. In cases like this, we say there is no solution to the equation. This means that there is no value for the variable that will make the equation true.
Great, looking at this problem, what we're going to be looking for is what value for x makes this equation true. Again that's what a solution means. So before I can start this problem I know I'm going to have to do some simplifying on both sides before I do any solving and solving is where you like add stuff to both sides, divide both sides by whatever. Okay what I mean by simplify is the distributing and combining like terms. So if I distribute that 3, 3x take away 6 plus 4, be really careful that the 3 gets multiplied by x and also by the -2 and then I can combine that -6 plus 4 is just the same thing as -2. Now that side of the equation is simplified.
Let's look at the other side 2x plus 6 plus x. Let's combine those Xs, 2x plus x is 3x plus 6. I'm just going to rewrite that one more time so that I can keep my work going and I can know where I am.
Now I have both sides of the equation that are simplified. What I would want to do next is get all of my Xs together on the same side of the equation but look out my friends, something weird is about to happen. The correct Mathy thing to do would be to subtract 3x from both sides but look I'm getting zero Xs, like it cancels out on both sides and what I have is -2 equals 6. This is really weird and what this means is that this equation has no solution. This is like your answer. You write no solution on your paper. What this means is there's no number fraction, decimal, upside down, whatever you wanted, like there's nothing that I could stick in there for x that would make this equation true and the way I know is because my Xs cancel out -2 is never equal to 6. That's how I know there's no solution. That's one of the weird things about Math is that almost always you get an answer, you get a solution. This is a situation where there's like no answer. You wouldn't have known that looking from the very beginning. This is another one of those problems you just need to work through carefully, show all your work and when you get something like this, don't freak out. Sometimes it happens, sometimes there's no solution.