2.1 – Linear Equations in One Variable

Objective – Solve linear equations using properties of equality.

-Solve linear equations that can be simplified by combining like terms.

-Solve linear equations involving fractions

Why is this important?

Linear Equations

Linear Equations in one variable – A linear equation in one variable is an equation that can

be written in the form ax + b = c

Where a, b, and c are real numbers and a 0 The Addition and Multiplication Properties of Equality

If a, b, and c, are real numbers, then a=b and a+c = b – c are equivalent equations Also a =b and ac = bc are equivalent equations as

long as c 0

Solve: Addition property of equality

2 x + 5 = 9

2x + 5 -5 = 9 -5

2x = 4

2x = 4

2 2

x = 2

Subtract 5 from both sides

Simplify

Divide both sides by 2

Simplify

Check

To see that 2 is the solution, replace x in the original equation with 2.

2 x + 5 = 9

2 (2) + 5 = 9

4 + 5 = 9

9 = 9

Give it a try

3 x + 6 = 12

3x + 6 – 6 = 12 –6

3x = 6

3x = 6

3 3

x = 2

Check

0.6 = 2 – 3.5c

0.6 = 2 – 3.5 (0.4)

0.6 = 2 – 1.4

0.6 = 0.6

Give it a try!

4.5 = 3 + 2.5 x

4.5 – 3 = 3 + 2.5 x – 3

1.5 = 2.5 x

1.5 = 2.5 x

2.5 2.5

0.6 = x

Solve – Combining like terms

-6 x – 1 + 5x = 3

-6x – 1 + 5x = 3

-x – 1 = 3

-x – 1 + 1 = 3 + 1

-x = 4

-1 -1

x = -4

Give it a try!

-2x + 2 – 4x = 20

-2x + 2 – 4x = 20

-6x + 2 = 20

-6x + 2 – 2 = 20 – 2

-6x = 18

-6 -6

x = -3

Solve: Distributive Property

2 (x – 3) = 5 x – 9

2(x – 3) = 5 x – 9

2x – 6 = 5 x – 9

2x – 6 – 5x = 5x –9 – 5x

-3 x – 6 = - 9

-3 x – 6 + 6 = - 9 + 6

-3x = -3

-3 -3

x = 1

Give it a try!

4 (x – 2) = 6x –10

4x – 8 = 6x – 10

2 = 2 x

1 = x

Solve: Adding/Subtracting Fractions

13 4 6y y

13 4 6

12 12y y

12 12 23 4y y

4 3 2y y

2y

Give it a try!

16 8 8x x

16 8

24 24 248

x x

4 3 3x x

3x

Solve: Multiplying Fractions5 31

22 2 8

x xx

5 312

2 2 88 8

x xx

38 8 8 8

5 12

2 2 8

xxx

4 5 4 16 3x x x 4 20 4 16 3x x x 4 24 15 3x x

11 24 3x 11 21x

2111

x

Give it a try!1 2 32

3 3 9

x xx

Solve: Decimals

0.3x + 0.1 = 0.27 x – 0.02

100(0.3x +0.1) = 100 (0.27x – 0.02)

100(0.3x) + 100 (0.1) = 100(0.27x) – 100(0.02)

30 x + 10 = 27 x – 2

30 x – 27 x = -2 –10

3 x = -12

3x = -12

3 3

x = - 4

Give it a try!

0.2x +0.1 = 0.12x – 0.06

100 (0.2 x + 0.1) = 100 (0.12 x – 0.06)

20 x + 10 = 12 x – 6

20 x – 12 x = -6 – 10

8 x = -16

x = -2

Solve: Contradiction

3x + 5 = 3(x+2)

3 x + 5 = 3 (x + 2)

3 x + 5 = 3x + 6

3 x + 5 – 3 x = 3 x + 6 – 3 x

5 = 6

This is a false statement…The original equation has no solution. Its solution set is written either by { } or

O. This equation is a contradiction.

Give it a try!

5 x – 1 = 5(x+3)

5 x – 1 = 5 x + 15

-1 = 15

{ }

Solve: Identity

6 x – 4 = 2 + 6 (x –1)

6x – 4 = 2 + 6x – 6

6 x – 4 = 6 x – 4

{x/ x is a real number}

6 x – 4 + 4 – 6x – 4 + 4

6 x = 6 x

6x – 6x = 6 x – 6x

0 = 0

This equation is called an identity!

Give it a try!

-4(x – 1) = -4x –9 +13

-4x + 4 = -4x + 4

{x / x is a real number}

Put it in words! Solving a linear equation in one variable

Step 1: Clear the equation of fractions by multiplying both sides of the equation by the LCD

Step 2: Use distributive property to remove grouping symbols such as parentheses.

Step 3: Combine like terms. Step 4: Isolate the variable by adding, subtracting,

multiplying, dividing (equality properties) Step 5: Check the solution in the original equation.