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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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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 !!!

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