# Simplifying & Solving Conditional Equations

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Many equations are conditional equations. This lesson will define conditional equations and go over how to simplify and solve simple conditional equations using various rules.

## Conditional Equations

Hannah just got \$5 for her allowance. That means candy money! She heads to the store and finds that her favorite candies are \$0.50 per piece. She's trying to figure out how many pieces she can get with her allowance. She lets the number of candies she will buy be x. Since each piece of candy costs \$0.50 and she is buying x candies, her total cost will by 0.50x. This total cost will have to be equal to \$5. She uses this rationale to set up the following equation:

• 0.5x = 5

If we solve this equation, we would find that the solution, or the value of the variable that makes the equation true, is x = 10, so Hannah can get 10 of her favorite candies.

In mathematics, this is an example of a conditional equation. A conditional equation is an equation that is true for some value or values of the variable, but not true for other values of the variable. In Hannah's case, we have that the equation is true for 10 but is not true for other values of x, such as 1. Therefore, the equation is a conditional equation.

The process of simplifying and solving conditional equations varies depending on the type of equation we are working with. It would be impossible to cover all the types of equations in one lesson, so we will stick to simple linear equations in this lesson. Let's take a look at simplifying and solving conditional equations!

## Simplifying and Solving Conditional Equations

When it comes to solving conditional equations, it all revolves around one goal - to isolate the variable on one side of the equation. Consider Hannah's equation again. This is a linear conditional equation. To solve linear conditional equations, we can isolate the variable by simplifying the equation using the following rules:

1. We can simplify both sides as much as possible.
2. We can add or subtract the same number or term from both sides.
3. We can multiply or divide the same number or term, aside from 0, on both sides.
4. We can interchange sides of the equation.

These rules will not change the value of the variable in the equation. In Hannah's equation, to isolate x, we need to get rid of the 0.5 that we are multiplying it by. If we divide 0.5x by 0.5, the 0.5 will cancel out, and x will be all by itself. By our rules, we can divide both sides of the equation by 0.5 without changing the value of the variable. Therefore, to solve the equation, we divide both sides by 0.5 and then simplify both sides as much as possible to isolate x.

As expected, we get x = 10. This is a very simple example of simplifying and solving conditional equations. Depending on the equation, isolating the variable can be fairly easy, like we just saw, or it can be extremely difficult. Let's consider another example that we can use our rules to solve.

## Example

Consider the following equation.

• 3x + 8 = x - 4

Let's verify that the equation is conditional by finding values of the variable that make the equation true and values that make the equation false. The easy part is finding a value of the variable that makes the equation false. Can you find any?

What if the variable is 0? Then we have the following:

• 3(0) + 8 = 0 - 4 or 8 = -4

Obviously, 8 ≠ -4, so x = 0 makes the equation false. There are many values that make the equation false, but all we need is one, and we've found that in x = 0. Now let's solve the equation using our rules to find a value of the variable that makes the equation true.

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