# Solving Linear Equations

## Solve a Formula for a Specific Variable


### Solve a Formula for a Specific Variable

We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.
It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.
Geometric formulas often need to be solved for another variable, too. The formula  is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.
In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.
The next example uses the formula for the surface area of a right cylinder.
Sometimes we might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.

### Use Formulas to Solve Geometry Applications

In this objective we will use some common geometry formulas. We will adapt our problem solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve.
In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.
When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.
The next example involves the area of a triangle. The area of a triangle is one-half the base times the height. We can write this as  where b = length of the base and h = height.
In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is  We can write this as a formula:  Also, since the triangle is a right triangle, we remember that a right triangle has one  angle.
Here, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.
The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.
We will use the Pythagorean Theorem in the next example.
The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as  where L is the length and  is the width. To solve the example, we will need to define the length in terms of the width.
The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as  where a, b, and c are the lengths of the sides.
Applications of these geometric properties can be found in many everyday situations as shown in the next example.

### Key Concepts

1. How To Solve Geometry Applications


2. The Pythagorean Theorem



### Practice Makes Perfect

Solve a Formula for a Specific Variable
In the following exercises, solve the given formula for the specified variable.
In the following exercises, solve for the formula for y.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth if necessary.
In the following exercises, solve using a geometry formula.
In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?