MTH 10905Algebra

SOLVING LINEAR EQUATIONS WITH A VARIABLE ON ONLY ONE SIDE OF

THE EQUATIONS

CHAPTER 2 SECTION 4

Solving Linear Equations

Solving Linear Equations in the form ax + b = c with a variable on only one side of the equal sign.

The general procedure is to “Isolate the variable”

Steps1. If the equation contains a fraction then multiply both sides by the LCD

to eliminate the fraction.2. Use the distributive property to remove any parentheses. 3. Combine any like terms that are on the same side of the equal sign.4. Use the addition property giving you an equation in the form of ax = b5. Use the multiplication property giving you x = or 1x =6. Check your answer

ba

ba

Linear Equations - Solve

Exp: 3 7 11

3 7 7 11 7

3 18

3

x

x

x

3x 18

3

18 3

6

x

x

No fractionsNo parentheses Use the addition property

Use the multiplication property

Check:

3x – 7 11(3)(6) – 7 11 18 – 7 11 11 = 11

???

Linear Equations - Solve

Exp #30: No fractions - No parentheses Use the addition property

Use the multiplication property

Check:

25 + 4x = 19 25 + 4(-3/2) 19 25 + -12/2 19 25 - 6 19 19 = 19

?

??

23

4664

2519419425

x

x

xx

x

Linear Equations - Solve

Exp: -3 4 2

3 4 4 2 4

-3 2

3

x

x

x

3x

23

2 3

x

No fractionsNo parentheses Use the addition property

Use the multiplication property

Check: -3x – 4 -2 -(3)(-2/3) – 4 -2 6/3 – 4 -2 2 – 4 -2 -2 = -2

????

Linear Equations - Solve

Exp #20:No fractions - No parentheses Use the addition property

Use the multiplication property

Check: 6 – 3x = 18 6 – 3(-4) 18 6 + 12 18 18 = 18

??

4 3

12

123 6183

1836

x

x

xxx

Linear Equations - Solve

Exp: 15 3 9

15 2 9

15 9 2 9 9

6 2

6 2 2

x x

x

x

x

2x

3 x

No fractionsNo parenthesesCombine like terms Use the addition property

Use the multiplication property

Check: 15 = 3x + 9 – x15 (3)(3) + 9 - 3

15 9 + 9 – 315 18 – 3 15 = 15

???

Linear Equations - Solve

Use the addition property first. If we use the multiplication property first we may or may not get the correct answer and we will usually have to do more work when working with fractions.

Exp: 4x – 7 (x + 3) = 2 CHECK4x + (-7)(x) + (-7)(3) = 2 4( ) – 7( ) + (-7)(3) = 2 4x -7x – 21 = 2 -3x – 21 = 2 + - 21 = 2 -3x = 2 + 21 -3x = 23 - 21 = 2 x =

23 - 21 = 2 2 = 2

23 23 or -3 3

23 -3

23 -3

92 -3

161 3

693

Linear Equations - Solve

Exp: CHECK 5x – (4x + 3) = 8 5x + (-1)(4x) + (-1)(3) = 8 5x –

(4x + 3) = 8 5x – 4x – 3 = 8 5(11) – ((4)(11) +

3) = 8 x – 3 = 8 55 – (44 + 3) = 8 x = 8 + 3 55 – 47 = 8 x = 11 8 = 8

Solving Linear Equations

Solving Linear Equations containing decimal numbers.

EXP: CHECK:2.36 0.05 5.21

0.05 2.36 5.21

.95 2.36 5.21

.95 5.21 2.36

.95 2.85

2.85 .95

3

x x

x x

x

x

x

x

x

2.36 0.05 5.21

3 2.36 (0.05)(3) ? 5.21

5.36 0.15 ? 5.21

5.21 5.21

x x

Solving Linear Equations

Solving Linear Equations containing decimal numbers.

EXP # 39: CHECK: 2.3x – 9.34 = 6.3

2.3x – 9.34 = 6.3 2.3(6.8) – 9.34 6.32.3x = 6.3 + 9.34 15.64 – 9.34 6.32.3x = 15.65 6.3 = 6.3x = 15.65/2.3x ≈ 6.8

??

Solving Linear Equations

Solving Linear Equations containing fractions.

When working with fractions you must find the LCD and multiply both sides of the equal side by the LCD.

Solving Linear Equations

EXP: 1 1 1 ( 3) 2 ( ) (3) 24 4 4

3 3 2 4 8 2 2 LCD = 4 4 4 4 4 1 4 4

8 3 8 3 4 4 4 4 4

5 5 (4) (4)4 4 4 4

5

x x

x x

x x

x x

x

Solving Linear Equations

EXP: 1 1 4 11 4 11 3 4 11 3 3 3

3312 33 11 33 -11

3

y y y y y y

y y y y

y

Solving Linear Equations

EXP:

1 2 1 1 2 1 LCD= 35 355 7 6 5 7 6

35 70 35 35 35 7 10 3 5 7 6 6 6

-3x 35 1 35 -3 6 3 18

x x x x

x x x x x

x

Overview

Evaluate – find numerical value

Simplify – perform operation and combine like terms

Solve – find the values of the variables

Check – Substitute the value back into the original equation

HOMEWORK 2.4

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