# Equations and Inequalities

### Topics

## Introduction to :

When an equation has more than one variable, we can't just say "solve the equation." That's like telling you to "name the One Tenor." You can't do that, because there are three of them. Not that many people could name the Three Tenors, either, but you get the point. #carrerasistheringoofthetenors

We need to specify which variable we want to get all by itself at the end. Rearranging an equation so that *x* (or *w*, or *y*, etc.) is all by itself is referred to as solving the equation for *x* (or *w*, or *y*, etc.) You may think this step will give you a headache, but just try following the plot of an opera sung entirely in Italian.

To solve an equation for a particular variable, we can perform the same actions we did when solving equations that only had one variable. We can add the same number to both sides, multiply both sides by the same number, etc.

The end goal is still to get a variable all by itself on one side of the = sign, but now the variable needs to be specified since there are several to choose from. You're like a mathematician in a candy store, except *all the candy is made of variables*.

### Sample Problem

Solve the equation *xy* = 3*z* + 2 for *z*.

Subtract 2 from both sides of the equation to find that

*xy* - 2 = 3*z*

and then divide both sides by 3 to find that

Since a formula is an equation that expresses one variable in terms of other variables, and life has lots of formulas, we often want to solve a formula for a particular variable. You may not have come across a ton of these life formulas yet, but you will. There are a lot of inequalities out there.

### Sample Problem

Let *A* be the area of a rectangle, l its length, and *w* its width. Then, the formula *A* = *lw *expresses the area of the rectangle in terms of its length and width. Hey, if you're trying to figure out the area of a rectangular desk to see if it will fit well in your new room, this could be one of those very life formulas we mentioned. Solve this formula for *w*.

This is a one-step problem. We divide both sides of the formula by* l*, write down and we are ready to roll.

Equations with multiple variables lend themselves to problems that ask you to find the value of some variable given the values of some other variable(s). Sometimes these are straightforward and require using a formula. See how we made a formula seem like a *good* thing?

### Sample Problem

Suppose a rectangle has area 30 cm^{2} and length 6 cm. What is the width of the rectangle?

We use the same formula, now substituting 30 for *A* and 6 for *l*, to find that

30 = 6*w*

This is an equation with only one variable: *w*. We know how to do this problem. We eat single variables for breakfast. (Let's be honest—we eat Sugar-O's.) We divide both sides by 6 to find that *w* = 5 cm.

When given area and length, finding width isn't so bad. We write down the formula for area, substitute the given numbers for area and length to get an equation with one variable, and solve the equation. However, if we were asked to find the widths of two hundred different rectangles, we would get tired of solving similar equations over and over and over again. Hopefully, you're not thinking of setting up two hundred desks in your new apartment.

But, even if you are, there is an easier way. Earlier, we solved the formula *A* = *lw* for *w* to find the new formula

We can use this formula from now on to find widths of rectangles.

#### Example 1

Suppose a rectangle has length 5 cm and width 2 cm. What is the area of the rectangle? |

#### Example 2

Find the width of a rectangle with length 3 in and area 45 in |

#### Example 3

Solve the formula |

#### Example 4

If the circumference of a circle is 40π, what is the radius of the circle? |

#### Example 5

If 5 |

#### Exercise 1

The volume *V* of a rectangular box is given by the formula *V* = *lwh* where *l* is length, *w* width, and *h* height. What is the height of a rectangular box with volume 56 cm^{3}, length 4 cm, and width 2 cm?

#### Exercise 2

Solve the formula 8*x* + 4*y* = 12 for *y*.

#### Exercise 3

Solve the formula *z*^{3} - 3*x* + *z*^{2} = *z* for *x*.

#### Exercise 4

If 4*x* - *y* = 5, find *x* when *y* = 15.

#### Exercise 5

If *y*^{2} + 2*y* - 3*x* = 5, find *x* when *y* = 2.