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Solving Equations with a Variable on Both Sides - Problem 2
In equations involving fractions, the same rules that apply to whole numbers apply to fractions. 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. Make sure your final answer is in simplest form.
This is the kind of problem that makes me nervous because of all the fractions but don't worry guys, you can still do these kinds of problems, you can still solve for x just make sure you keep in mind the same processes you would use if these were whole numbers.
First thing I want to do is distribute this 1/2 so I'll have 1/2x, what's 1/2 of -8? Well let's see, 1/2 of 8 is 4, so I'm going to have -4 when I account that negative sign. There's that sign -2 minus 5/2x. Next thing I want to do is combine all of my Xs and combine all my regular numbers or constants. This is where your solving techniques come handy, like for example I have -4 and -2 that are on opposite sides of the equals sign. I don't want them to be on opposite sides, I want them to be together so I'm going to go through and add 4 to both sides. So now I'll have 1/2x equals 2 take away 5/2x.
By the way adding 4 is just one possible step at this point, there's lots of different ways you could proceed. If you wanted to you could subtract 1/2x or you could add 5/2x, you can do whatever you want to that's like the opposite of one of these operations. I just chose this, there was really no reason why. Okay so now I have my regular numbers are combined to just that 2, I need to get this x over here so that all my x terms are combined. I want to add 5/2 x to both sides, 5 halves of x, I'm not sure how to say that. 1/2x plus 5/2x is going to be 6/2 or you can probably do in your brain 6/2 if you reduce that fraction that's just going to give me 3x. Let me say that one more time. I had 1/2x, I added 5 halves more so it's 6 halves x, 6 halves reduced to 3. That's how I got that.
Okay I'm almost done, 3 times what number gives me 2? I'm not totally sure because it's not a whole integer but when you write it out like this you can see that x is going to be equal to 2/3. Fractional answer, it's okay, sometimes you don't get a whole number as your answer. It's totally normal.
Keep in mind with fractions you do the same processes you would do if you had whole numbers. Like in this problem I would have done the same process if this had been like 3 and 8. You would do the same stuff, it's just that now I have to deal with fractions. It's one extra thing to keep your brain sharp.