Linear Equations

Vocabulary:

An equation in x is a statement that two algebraic expressions are equal.

To solve –> finding all the values of x for which the equation is true.

The values of x are called solutions.

identity –> true for every real number in the domain

conditional equation –> is true for just some or even none

Linear Equation

A linear equation in one variable x is an equation that can be written in the standard form

ax+b=0  where a and b are real numbers with a\neq 0 .

To solve linear equations:

1.  Simplify both sides [remove brackets and combine like terms]

2.  Isolate x on one side of the equation by ‘undo’ ing BEDMAS. 

  1. AS –> add & subtract first

  2. DM –> multiply or divide last

Extraneous Solution –> sometimes a solution will not work. 

Ex.  a) ‘D’ – denominator = 0

        b) negative number under the root sign

Examples:    a)\; 2\left ( x-1 \right )=2x-2

                               2x-2=2x-2

                                      LS = RS       Identity

                      b)\; -6\left ( x-3 \right )+5=-2x+10

                              -6x+18+5=-2x+10

                                     -6x+23=-2x+10

                                                13=4x

                                                  x=\frac{13}{4}                Conditional Equation

                     c)\; x-3\left ( 2x+3 \right )=8-5x

                               x-6x-9=8-5x

                                   -5x-9=8-5x

                                            0x=17

                                              0=17   this is false  0\neq 17  so no solution.

                     d)\; 10-\frac{13}{x}=4+\frac{5}{x}

                        \frac{10x-13}{x}=\frac{4x+5}{x}

                        10x-13=4x+5

                                  6x=18

                                    x=3 

                     e)\; \frac{x}{x+4}+\frac{4}{x+4}+2=0

                \left ( x+4 \right )\left ( \frac{x}{x+4}+\frac{4}{x+4}+2=0\right )

                                    x+4+2x+8=0

                                                3x+12=0

                                                         3x=-12

                                                           x=-4  Extraneous Root

Therefore there is no solution. x=-4 makes the denominator = 0.

To review the concepts presented in this post, watch this free video about solving a linear equation.

Filed under: Pre-requisites for Pre-Calculus

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