Section 1.1 - 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

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

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.

Examples

• [1] Solve for $x$: $\quad \displaystyle -2 = \frac{3x+11}{2}$ (Solution)

• [2] Solve for $v$: $\quad \displaystyle -6v-21 = 5(v-2)$ (Solution)

• [3] Solve for $x: \quad \displaystyle -5(4x-4)+7x=3(x+4)$ (Solution)

• [4] Solve for $x: \quad \displaystyle \frac{9x}{8}-2=\frac{11x}{10}$ (Solution)

• [5] Solve for $y: \quad \displaystyle -\frac{1}{8}y-\frac{5}{4}=-\frac{1}{3}$ (Solution)

• [6] Solve for $v: \quad \displaystyle 7v-\frac{1}{5}=-\frac{5}{2}v-\frac{6}{5}$ (Solution)

• [7] Solve for $x: \quad \displaystyle 5=\frac{3x+3}{4}+\frac{5x-1}{7}$ (Solution)

• [8] Solve for $w: \quad \displaystyle 2(w+2)+w=3(w-1)+6$ (Solution)

• [9] Solve for $w: \quad \displaystyle -6(w+1)+8w=2(w-3)$ (Solution)

Application Problems

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

1. Understand the problem.

   • Read and reread 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.

   • Solve your equation.

4. Look back.

   • Give a complete answer.
   • 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.

Examples

• [10] Write an algebraic equation that represents the following problem situation: Four more than the quotient of a number and 6 is 9. (Solution)

• [11] 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.

1. Write an algebraic expression for the total number of calories burned by the two of them.
2. Together they burn a total of 505.6 calories. How far did each person walk?

(Solution)

For further help...

1. Khan Academy has excellent videos on linear equations and applications. Start at this page.
2. Do a Google search for "solving one-variable linear equations".
3. Do a Google search for "word problems with linear equations".