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Simplifying & Solving Equivalent Equations

Instructor: Michelle Vannoy
When would two equations for one unknown variable be equal? Why would this information be useful? How would you solve for that variable and what would that variable mean? This lesson will explain all of these questions.

When Are Equivalent Equations Used?

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 $6.00 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. 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 the story above.

How Would You Set Up and Solve These Types of Equations?

Let's continue our example above.

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: 6x + 12 = Total number of shirts bought

T-Shirts For Less: 7.50x +9 = Total number of 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: 6x + 12 = 7.50x +9. Solving for x tells you how many shirts would cost the same amount, no matter which company you order from.

How to Solve an Equation With Variables on Both Sides

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.

Step1 - Check for 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 (whichever is opposite of the current operation) the variable with its coefficient from both sides of the equation. By subtracting 7.50x (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. Accomplish this by adding or subtracting (whichever is the opposite of the current operation) the constant from both sides of the equation. The 12 is a positive 12 so we do the opposite operation which would be subtracting 12. What you do to one side of the equation you must do to the other.

Step 5 - Isolate the variable by multiplying or dividing the coefficient (whichever is opposite of the current operation) of the variable on both sides of the equation. At this point, you should have the variable equals 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.

Let's work through our example problem together:

Solving equivalent equation example
Solving Equivalent Equations

What Does It Mean?

Your 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.

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