# Section 1.1 & 1.2 - Linear Equations

Section Objectives

1. Recognize a single-variable equation as linear and solve it (0, 1, or inf. many soln's).
2. Translate a problem situation into an equation and solve.

A linear equation is an equation in which each variable appears only with an exponent of 1 and not in the denominator of a fraction nor in a radical.

For now, we will focus on linear equations of one variable. Every linear equation of one variable, $x$, can be written in the form $ax+b=0$.

### Solving Linear Equations

To solve an equation means to find ALL replacements for the variable that make the equation true. We typically solve linear equations by constructing a sequence of simpler, equivalent equations. At some point in this process, the solution becomes obvious.

#### For example...

Solve for $x$: $3(x-1)+x=-x+7$
$3x-3+x=-x+7$
$4x-3=-x+7$
$5x-3=7$
$5x=10$
$x=2$

#### General Guidelines for Solving Linear Equations

• Remove parentheses (if necessary) by using the distributive property.
• Combine like terms on each side of the equation.
• Use addition or subtraction to move all variables to one side of the equation and all constants to the other.
• If possible, use multiplication or division to get an equation of the form $x=$ constant.
• If the step above is not possible, your equation is either an identity (every number is a sol'n) or a contradiction (no number is a sol'n).

Important note: At any step in the process above, it could be helpful to clear fractions by multiplying both sides of the equation by the LCM of all denominators.

### Application Problems

In mathematics, there is a well-known, very broad, four-step problem solving process:

1. Understand the problem.

• What is the problem asking for?
• What information is given?
• Assign variables to unspecified quantities.
• Sketch any helpful tables or diagrams.
2. Devise a plan.

• Translate the problem situation into an equation.
3. Carry out the plan.

4. Look back.

• Check your solution in the original wording of the problem.

Steps 1 & 2 involve defining variables and translating words to equations. We'll do examples in class, but this sheet may help you with your translation skills.

#### Example

When walking, Jose burns 96 calories per mile and Sara burns 64 calories per mile. One day the two of them walk a total of 7 miles. Let $x$ represent the number of miles walked by Sara.

a. Write an algebraic expression for the total number of calories burned by the two of them.

b. Together they burn a total of 505.6 calories. How far did each person walk?