Solving equations can seem impossible sometimes. How do you feel when you see a problem like this:
5x + 3x * 10 = 10x + 50. Solve for x.
If you can answer this problem easily, that’s great! If you can’t, that’s okay. Hopefully this article will help you feel more comfortable answering similar questions.
The thing about equations is that each side of the equal sign represents the same value. For instance, 1 + 1 = 2, so 1 + 1 has the same value as 2. Let’s try a simple example: If x + 2 = 10, then you know that if you add 2 to x, you get 10. So what’s x? x = 10 – 2. So x = 8. Now check: 8 + 2 = 10? Correct! Not so bad, right?
Most problems like the one above can be done in a few simple steps. (Note: Opposite operations: add & subtract, multiply & divide, root & exponent)
1. Get x on one side of the equation. How do we get all of the xs to one side? We use opposite operations to move the xs. And remember, because it’s an equation, whatever we do to one side, we do to the other! We’ll keep moving the xs one step at a time until they are all on one side, then we can move to the next step.
2. Simplify & combine like terms. Like terms have a) the same base and b) the same exponent. We must follow the order of operations (remember PEMDAS?) when simplifying. And remember the properties of real numbers (identity, associative, commutative, and distributive). We’ll keep combining xs until there’s only one left. Then, we can move to the next step.
3. Solve. Use the order of operations to get x alone and use the opposite operation to solve for x.
4. Check. Now plug in your answer for x, and check to make sure both sides are equal.
If they’re not equal, then go back and try again. This is an important step! Many times people make little errors and don’t ever realize it. If you check, then you’ll know if your answer is right.
So here is the final worked out solution:
5x + 3x * 10 = 10x + 50 1. Get x on one side of the equation.
-10x -10x
5x + 3x * 10 – 10x = 50 2. Simplify. Multiply 3x * 10 = 3 * 10 * x = 30x
5x + 30x – 10x = 50 2. Simplify. Add 5x + 30x = 35x
35x – 10x = 50 2. Simplify. Subtract 35x – 10x = 25x
25x = 50 3. Solve. Divide 50 / 25 = 2
x = 2
5(2) + 3(2) * 10 = 10(2) + 50 ? 4. Check.
10 + 6 * 10 = 20 + 50 ?
10 + 60 = 20 + 50 ?
70 = 70
