Suppose you were traveling on a paddle boat at a constant speed. In 6 minutes, you traveled meters, and in 10 minutes, you traveled meters. Could you find the value of in this scenario? If so, how would you do it?

### Rational Equations Using Proportions

You are now ready to solve rational equations! There are two main methods you will learn to solve rational equations:

- Cross-products
- Lowest common denominators

This Concept will focus on cross-products. When two rational expressions are equal, a **proportion** is created and can be solved using its cross-products.

For example, to solve , cross multiply and set the products to be equal.

Solve for :

Let's solve the following rational equations:

- .

Notice that this equation has a degree of two; that is, it is a quadratic equation. We can solve it using the quadratic formula.

Start by cross multiplying:

Since this equation has a squared term as its highest power, it is a quadratic equation. We can solve this by using the quadratic formula, or by factoring.

Since there are no common factors, start by finding the product of the coefficient in front of the squared term and the constant:

What factors of -6 add up to 5?

That would be -6 and 1, since -6+1=-5

Factor, beginning by breaking up the middle term, , as above:

Use the Zero Product Principle:

### Examples

#### Example 1

Earlier, were asked to find the value of given that you were traveling on a paddle boat at a constant speed. You traveled meters in 6 minutes, and meters in 10 minutes.

You can use cross-products to find the value of in this scenario.

First, let's set up an equation using the given information. The distance traveled in a specific amount of time can be written as a fraction of the distance over the time.

Now, we can cross multiply.

#### Example 2

Solve* .*

### Review

Solve the following equations.

**Mixed Review**

- Divide: .
- Solve for .
- Find the discriminant of and determine the nature of the roots.
- Simplify .
- Simplify .
- Divide: .
- Simplify .

### Review (Answers)

To see the Review answers, open this PDF file and look for section 12.8.