Solving Linear Equations

Introduction

An equation is a statement that two mathematical expressions are equal. 2x + 4 = 8 is an equation.

To solve an equation involving x means you are trying to find all the values of x that make the equation true.

The solution set is the set of all real numbers that are solutions to the equation.

If every real number solves an equation, the equation is called an identity. For example, 2(x- 1) = 2x - 2 is an identity equation because any real number you substitute for x makes this equation true.

If only specific real numbers solve an equation, the equation is called a conditional equation. For example,

x + 4 = 6 is true if and only if x = 2 so it is a conditional equation.

An equation that has no solution is said to have an empty set solution. For example, + 1 = 0 has no real number solutions. We will use to indicate the empty set.

You check a solution set by merely substituting your answer in the original equation to verify your results.

Example: Solve x + 4 = 6

Answer x = 2

To check: Substitute 2 everywhere you have an x.

And the solution checks out fine!

Trial Solutions

See if 5 satisfies the equation 2x + 3 = 10

To check 5 in the equation, substitute it wherever you have an x.

2x + 3 = 10 Original equation

Substitute 5 for x

So 5 is not a solution to this equation!!!!

Two equations that have the same solution set are called equivalent equations.

To solve linear equations, you are trying to work your way to the simple equation

x = ______

1. Remove any parentheses in the equation.
2. Then try to get all the unknown terms (those involving x ) on one side of the equation and all the constant values on the other side. I always put the x term on the left hand side of the equation and the constant terms on the right hand side of the equation. Remember - you can do "almost" anything to an equation as long as you do it to both sides of the equation!! That means you can add the same quantity to both sides; you can subtract the same quantity from both sides; you can multiply both sides of an equation through by the same number; and you can divide both sides of an equation through by the same number.
3. Collect like terms
4. And remember, it is okay to completely swap each side of the equation. 2 = x is the same as x = 2 !!!

A linear equation in one variable x is an equation that can be written in the standard form ax + b = c. It is called a first-degree equation because its variable has an exponent of 1. A linear equation should always have exactly one solution.

Examples

Solve the following standard linear equation for x and check your answer by substitution.

3x - 7 = 2
3x -7 + 7 = 2 + 7	Add 7 to both sides
3x = 9	Collect like terms
	Now divide both sides by 3
x = 3	This is your answer

CHECK: Substitute 3 for each x in the original equation

and the solution checks !!!