## Equivalent Equations

You are planning a family reunion and have decided to buy custom-printed shirts for everyone to wear. Two companies have been equally recommended by your friends. The first company, Too Cool Tees, charges $6 per shirt and $12 processing and handling fee. The second company, T-Shirts For Less, costs $7.50 per shirt and $9 processing and handling fee. You are not sure how many people are going to be at the family reunion. You ask yourself, at what point will the cost be the same from either company? Equivalent equations will answer this question.

**Equivalent equations** are two equations that have the same solution. They are used anytime multiple equations with the same variable need to equal each other, just like in our example.

## Setting Up to Solve

Let *x* equal the number of t-shirts ordered.

The equation for each of the two companies for the cost of the shirt plus processing and handling fees would be:

Too Cool Tees: 6*x* + 12 = total shirts bought

T-Shirts For Less: 7.50*x* +9 = total shirts bought

Since you do not know how many family members are coming to the reunion yet and you want to see at what point the t-shirts from the two companies would cost the same, you would set the equations equal to each other and solve. Anytime a problem tells you to find when two equations are the same, this means to set the two equations equal to each other. Your problem for our story would look like this: 6*x* + 12 = 7.50*x* +9. Solving for *x* tells you how many shirts would cost the same amount, no matter which company you order from.

## Solving

To solve any multi-step equation, you need to do a set of five steps in order. Not all problems will have all steps present, but you should check for them in order anyways.

### Step 1

Check for the distributive property on each side of the equal sign. If present, complete the distributive property. There is no distributive property in our example.

### Step 2

Check for combining like terms on each side of the equal sign. If present, combine all like terms on the same sign of the equal sign. There are no like terms present in our example.

### Step 3

Collect all the variables on the same side of the equal sign. Do this by adding or subtracting the variable with its coefficient from both sides of the equation. Add or subtract based on whichever is opposite of the current operation. By subtracting 7.50*x* (subtracting is opposite of a positive 7.50*x*), we can move the *x*'s on the same side of the equation. What you do to one side of the equation you must do to the other side to keep the equation equal.

### Step 4

Collect all constants on the opposite side of the equation as the variable. You can do this by adding or subtracting the constant from both sides of the equation. Again, add or subtract based on whichever is the opposite of the current operation. The 12 is a positive 12 so we do the opposite operation, which would be subtracting 12.

### Step 5

Isolate the variable by multiplying or dividing, whichever is opposite of the current operation, the coefficient of the variable on both sides of the equation. At this point, you should have the variable equaling a number. The variable is being multiplied by -1.50, so the opposite operation is to divide by -1.50. This has to be done to both sides to keep the problem equal.

## Answer Meaning

The solution is *x* = 2. This means either company will cost you the same amount of money if you are ordering two shirts. You can substitute 2 in for *x* and find out exactly what you would be paying. 6(2) + 12 = 24. Since the companies cost the same at two shirts, you can determine that for each shirt after that T-Shirts For Less is going to cost you $1.50 more for each shirt ordered than if you ordered from Too Cool Tees.

## Uses

This type of information helps you to make quick decisions based on the data. In the story, if you know that there are going to be more than two shirts ordered and all other details, like quality and shipping time, are the same, you should order from Too Cool Tees. You no longer have to waste time calculating the cost from each company.

The point at which the value of *x* is equal is called the **break-even point**. Retail and product research and development departments use the break-even point to determine what price they would have to sell the product to break even and then add the percentage profit they want to make to the break-even price. Determining when two values are equal is very useful information.

## Lesson Summary

**Equivalent equations** are two equations set equal to each other so that the variable has the same value in each equation. They are used to find break-even points. The **break-even point** is the point at which the value of the variable is equal in equivalent equations. For example, the selling point in which a company is selling their product for the same amount that it cost them to make it. This point is important to a company if they want to make a profit

To solve equivalent equations, follow these five steps:

- Step 1 - Use the distributive property, if necessary.
- Step 2 - Combine like terms on the same side of the equal sign, if needed.
- Step 3 - Collect the variable being solved for on the same side of the equation.
- Step 4 - Collect all the constants on the opposite side of the equation.
- Step 5 - Multiply or divide by the coefficients on both sides.

Remember to use this method anytime you have two equations that need to equal each other or to find break-even points for a product.