LINEAR EQUATIONS

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Linear Equations

Definitions

  • Linear Equation - An equation involving variables of the first degree only (i.e., no quadratic (x2) or cubic (x3) terms).
    For example, 10x + 2 = 5x + 12 is a linear equation while x2 + 4 = x - 5 is not a linear equation but is rather a quadratic equation.
  • Solution - A number that, when substituted for the unknown, will make the equation true.
    For example, in the equation x + 3 = 4, the number 1 is the solution since 1 + 3 = 4

Examples of Linear and Non Linear Equations

Linear Equations

The following is a linear equation, even though it has two different variables.
5x + 7y = 14
The following is a system of linear equations, even though it has two different equations with two different variables in each equation.
10x + 15y = 72
22x + 3y = 102

Non Linear Equations

The following are not linear equations since they have an exponent with a power that is not 1.
x2 - 2x + 1 = 0
x-5 + y-5 + 15x2 + 2 = 0

Simple Linear Equations

You must perform the same operations on both sides of the equation.
A simple linear equation is one with a single variable. In this case, solving for the unknown variable is accomplished by isolating that unknown. When performing mathematical operations, remember that the same operation that is done to one side of the equation must be done to the other side of the equation. For example, if you add 5 to the left side of the equation, you must add 5 to the right side of the equation.
5x + 20 = 2x - 10
5x - 2x + 20 = 2x - 2x - 10
3x + 20 = - 10
3x + 20 - 20 = - 10 - 20
3x = - 30
x = -10

10x + 15 - 5x + 30 = 0
5x + 45 = 0
5x + 45 - 45 = 0 - 45
5x = -45
x = -9

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