How to Solve Problems with the Elimination in Algebra: Examples

Instructor: Joseph Vigil
In this lesson, discover how to solve algebraic equations by elimination and see a few examples of the process in action. Then test your new knowledge with a brief quiz.

The Situation

Summer's finally here, and you've found the perfect swimsuit for the pool parties sure to come. The only problem is it costs \$24.99, and you only have \$22.99.

Without even thinking about the process, you probably know rather quickly that you're \$2 short. But how did you do that? Let's take a closer look at how we can solve problems like this one.

Knowns and Unknowns

In the above situation, let's sort out the information we know and the information we need to know. We know:

2. You have \$22.99.

We need to know what amount of money, when added to what you already have, will give you \$24.99. Let's put that information into an equation.

\$22.99 (what you already have) + ? (how much more we need) = \$24.99 (the cost of the swimsuit)

Since we have an unknown value, we can use a variable, which is a letter we use to stand in for an unknown value. Let's use x, a very common variable, for the unknown value and take out the dollar signs and commentary. We now have

22.99 + x = 24.99

But, ideally, we want an equation that says x equals some number. In other words, we want to get x by itself on one side of the equation. How do we zap away that 22.99?

Elimination

This is where inverse operations come in handy. Inverse operations are simply opposite mathematical operations. For example, addition and subtraction are inverse operations of each other. Likewise, multiplication and division are also inverse operations because they basically undo each other.

Since we're adding x and 22.99 together, we'll need to use subtraction since it's addition's inverse operation. So let's subtract 22.99:

We've eliminated the 22.99 and now have x by itself. But we're not quite done yet. See that equal sign? The two sides of the equation aren't equal anymore because we changed one side. To keep both sides equal, we have to do the same operation on them both. So just like we subtracted 22.99 form the left side, we need to subtract 22.99 from the right side:

Now we've restored the balance between both sides of the equation. And we've found what x equals to! Since x = \$2.00, we need \$2 to buy that swimsuit.

We've used elimination to solve this equation.

Another Example

Your teacher will have twenty-five students this year. He knows he wants five rows of desks, but how many desks will need to go in each row?

Let's determine knowns and unknowns again. We know:

1. We'll have five rows of desks
2. We'll need twenty-five desks altogether

What we need to know is how many desks must go in each row for us to end up with twenty-five in all. Well, this is a grouping situation, so we're dealing with multiplication. So let's set up our equation:

5 (rows of desks) * ? = 25 (the total number of desks needed)

Let's substitute the variable x for our unknown value again:

5 * x = 25

Since we're multiplying here, we'll need to use division since it's the inverse operation of multiplication. And since we want to get the x by itself, we'll eliminate the 5 by dividing both sides (remember, we need to perform the same operation on both sides of the equation to keep them equal) by 5.

We've solved for x. In order for the teacher to have enough desks, he'll need 5 in each row.

More Complex Equations

Let's take a look at a more difficult equation. Sometimes, you'll see a variable on both sides of an equation. For example:

2x + 10 = x + 20

First, we'll need to get x on one side of the equation, which means we'll need to eliminate it from one side. It'll be easier to eliminate it from the right side, since there's only one x there. So let's subtract x from both sides:

Now, it's just a matter of eliminating the 10 to get x by itself:

In this equation, x = 10.

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