Solving Algebraic Equations: Definition & Examples

Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

If you shouted, 'Hey! Why are there letters in my math?' the first time you saw algebra, then you definitely want to check out this lesson where we'll learn why we use letters for mathematics. (Hint: it isn't to drive you crazy.)

Math with Letters

In your opinion, is it easier to write '3x + 2 = 17' or 'I am thinking of a number, but I'm not going to tell you what it is. I will tell you that if you add three of the numbers I am thinking of, and then add two to that total, you will end up with seventeen,' instead?

Hopefully you agree that the mathematical expression says the exact same thing in a quicker fashion. This is because mathematics was designed to express concepts and calculations in a much more concise fashion than other human languages.

Basic Terminology

Math with letters is really just an extension of math without letters. Algebra simply makes it easier to talk about something with an unknown value and not have to make a crazy statement like we did above.

Mathematicians have agreed to call the letter that is used to represent an unknown quantity a variable. Just to make things confusing, it's still called a variable even when it represents a single specific number, as is the case with our sample equation. 5 is the only number that makes the equality 3x + 2 = 17 true. But, even after you know that, the 'x' is still called a variable.

The '3' in 3x + 2 = 17 is called the coefficient, while the '2' and '17' are called constants and we can refer to them as constant terms. Any terms that are multiplied by the same variable or combination of variables are like terms. 3y and 10y are like terms, as are 3xy and 17.23xy. Compare those to 3x and 7y, which are not like terms and can't be combined.

Now that we got that out of the way let's get algebraic equations straight.

Definition of Algebraic Equation

There are a few rules we must observe:

  • an algebraic equation must contain a variable
  • the variable must be multiplied by a coefficient that is not zero
  • there should be an equal sign

Is our equation, 3x + 2 = 17, an algebraic equation?

Yes! It has a variable multiplied by a non-zero coefficient (3) and has an equal sign, so it meets our requirements.

Solving Single Variable Equations

'Solving' an algebraic equation just means manipulating the equation so that the variable is by itself on one side of the equation and everything else is on the other side of the equation. Once 'everything else' is simplified, the equation is solved.

The most simple algebraic equation you could have would be something like x=5, which is both an algebraic equation and its own solution.

Let's try something a little harder: y + 5 = 10

How can we get the y by itself? Why, get rid of the 5 of course! Only it's not quite that easy. Equation sides are a lot like siblings: if you do something for one and not the other, someone is going to start screaming, 'That's not fair!' To avoid that situation, whatever we do to one side of the equation we need to do to the other as well. What do we need to do the left side to get rid of that pesky 5?

Subtract 5 from both sides of the equation. Doing so makes our equation become:

y + 5 - 5 = 10 - 5

This is kind of clunky, so let's combine like terms.

y + (5 - 5) = (10 - 5)

5 - 5 = 0 and 10 - 5 = 5, so our equation becomes:

y = 5

This is now solved! As you get more familiar with these kinds of operations, you can skip the intermediate steps and just go from y + 5 = 10 to y = 5 in a single step. For now, though, you should write each step out. It's good practice, and also helps your teachers figure out which steps you're having trouble with.

Another tip: don't assume you know how much space you'll need to solve an equation. This often leads to a mess, so avoid it! Leave plenty of paper to work out each solution so you never run out of room. Better yet, don't write down anything for next problem until you've finished the one you are working on.

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