Martin-Gay, Beginning Algebra, 5ed 22

Martin-Gay, Beginning Algebra, 5ed 33

Using Both Properties

Divide both sides by 3.3

27

3

3z

Example:

3z – 1 = 26

3z = 27 Simplify both sides.

z = 9 Simplify both sides.

3z – 1 + 1 = 26 + 1 Add 1 to both sides.

Martin-Gay, Beginning Algebra, 5ed 44

Using Both Properties

Example:

12x + 30 + 8x – 6 = 10

20x + 24 = 10 Simplify the left side.

20x = – 14 Simplify both sides.

20

14

20

20 x

Divide both sides by 20.

20x + 24 + (– 24) = 10 + (– 24) Add –24 to both sides.

7

10x Simplify both sides.

Martin-Gay, Beginning Algebra, 5ed 55

Solving Linear Equations

301093 yy Simplify both sides.

21 7y Simplify both sides.

y 3 Divide both sides by 7.

Example: Solve the equation.3( 3) 2 6

5y y

Add –3y to both sides.30)3(109)3(3 yyyy

Add –30 to both sides.)30(307)30(9 y

Multiply both sides by 5. 3( 3)5 5 2 6

5

yy

Simplify both sides.9 7 30y

Martin-Gay, Beginning Algebra, 5ed 66

– 0.01(5a + 4) = 0.04 – 0.01(a + 4)

Multiply both sides by 100.

Example: Solve the equation.

– 1(5a + 4) = 4 – 1(a + 4)

Apply the distributive property. – 5a – 4 = 4 – a – 4

Add a to both sides and simplify. – 4a – 4 = 0

Add 4 to both sides and simplify. – 4a = 4

Divide both sides by -4. a = – 1

– 5a – 4 = – a Simplify both sides

Martin-Gay, Beginning Algebra, 5ed 77

– 0.01(5a + 4) = 0.04 – 0.01(a + 4)

Example: Solve the equation.

– 0.05a – 0.04 = 0.04 – 0.01a – 0.04 Apply the distributive property.

– 0.05a – 0.04 = – 0.01a

Add 0.01a to both sides. – 0.04a – 0.04 = 0

Add 0.04 to both sides and simplify. – 0.04a = 0.04

Divide both sides by -0.04. a = – 1

Simplify both sides.

Martin-Gay, Beginning Algebra, 5ed 88

Identity Equations

5x – 5 = 2(x + 1) + 3x – 7

5x – 5 = 2x + 2 + 3x – 7 Use distributive property.

5x – 5 = 5x – 5 Simplify the right side.

Both sides of the equation are identical. This equation will be true for every x that is substituted into the equation, the solution is “all real numbers.”

Example: Solve the equation.

0 = 0 Add 5 to both sides.

– 5 = – 5 Add -5x to both sides.

Identity Equation.

Martin-Gay, Beginning Algebra, 5ed 99

Contradiction Equations

Since no value for the variable x can be substituted into this equation that will make this a true statement, there is “no solution.”

3x – 7 = 3(x + 1)

3x – 7 = 3x + 3 Use distributive property.

– 7 = 3 Simplify both sides.

3x + (– 3x) – 7 = 3x + (– 3x) + 3 Add –3x to both sides.

Example: Solve the equation.

Contradiction Equation.

Martin-Gay, Beginning Algebra, 5ed 1010

Solving Linear Equations

Martin-Gay, Beginning Algebra, 5ed 1111

Examples

c)

a)

d)

b)