The Addition Principle of Equality2.22.2

1. Determine whether a given equation is linear.2. Solve linear equations in one variable using the

addition principle.3. Solve equations with variables on both sides of

the equal sign.4. Solve identities and contradictions.5. Solve application problems.

Linear equation: An equation in which each variable term contains a single variable raised to an exponent of 1.

Linear equation in one variable: An equation that can be written in the form ax + b = c, where a, b, and c are real numbers and a 0.

Addition Principle of EqualityAddition Principle of Equality

1253 x 5a 723 yx

7532 2 xxNonlinear equations: 1253 3 x

Addition Principle of EqualityAddition Principle of Equality

5 5= 5 5 + 2= 5 + 2 5 + 2=

We can add (or subtract) the same quantity to (from) both sides of an equation and still have a true statement.

Left side Right side

Addition Principle of EqualityAddition Principle of Equality

If a = b, then a + c = b + c for all real numbers a, b, and c.

To clear a term in an equation, add the additive inverse of that term to both sides of the equation.

5 –5

–3 3

2x –2x

Additive Inverse

–7x 7x

x + 15 is an expression, so we can’t use the addition principle of equality!!!!

x = 3 is the same as 3 = x.

Addition Principle of EqualityAddition Principle of Equality

x – 19 = –34 Additive Inverse:

+ 19 + 19x + 0 = –15

x = 15

Since we added 19 to the left side, we must add 19 to the right side as well.

Check: 19 34 x

15 19 34 34 34

Replace x with –15 .

True, so –15 is the solution.

Goal: x = some number

19

Addition Principle of EqualityAddition Principle of Equality

x + 7 = 20 Additive Inverse:

–7 –7x + 0 = 13

x = 13

Check:

True, so 13 is the solution.

Goal: x = some number

–7

x + 7 = 20

13 + 7 = 20

20 = 20

Addition Principle of EqualityAddition Principle of Equality

m + 5 = 12 – 4

Additive Inverse: –5 –5

m + 0 = 3

m = 3

Check:

True, so 3 is the solution.

Goal: m = some number

–5

Simplify sides if you can.m + 5 = 8

m + 5 = 12 – 4

3 + 5 = 12 – 4

8 = 8

To Solve Linear Equations1. Simplify both sides of the equation as needed. a. Distribute to clear parentheses. b. Combine like terms.2. Use the addition principle so that all variable terms

are on one side of the equation. 3. Use the addition principle so that all constants are

on the other side.

Addition Principle of EqualityAddition Principle of Equality

Addition Principle of EqualityAddition Principle of Equality

Solve: 4253 xx Additive Inverse: –2x

–2x –2x

4051 xx

45 x Additive Inverse: –5

–5 –510 x

1xCheck: 412513

4253

22 solution. the is 1x

Addition Principle of EqualityAddition Principle of Equality

Solve: 7324 xxx Simplify sides if you can.

732 xx Additive Inverse: –x–x –x

7031 x73 x Additive Inverse: –3

–3 –3

10xCheck:

7103102104

1732040 1717 solution. the is 10x

Addition Principle of EqualityAddition Principle of Equality

Solve: 1042 xx Simplify sides if you can.

1082 xx Additive Inverse: –x

–x –x

108 x Additive Inverse: +8+8 +8

18x Check: 10184182 1018142

2828

solution. the is 18x

Solve for t. – 8t + 4 +14t = – 4t + 3 + 9t

a) t = 5

b) t = 3

c) t = 1

d) t = –1

2.2

Solve for t. – 8t + 4 +14t = – 4t + 3 + 9t

a) t = 5

b) t = 3

c) t = 1

d) t = –1

2.2

Addition Principle of EqualityAddition Principle of Equality

Solve: 12243 xxx Simplify sides if you can.

122123 xxx123123 xx Additive Inverse: –3x

–3x –3x120120 xx

1212 Lost the variable term!!True statement.

IdentitySolution is ALL REAL NUMBERS.

Identity: Identity: An equation that has every real number as a solution.

Addition Principle of EqualityAddition Principle of Equality

Solve: xxx 1342 Simplify sides if you can.

Additive Inverse:

–2x –2x

xxx 33823282 xx –2x

3080 xx

Lost the variable term!!False statement.

ContradictionThere is NO SOLUTION.

Contradiction: Contradiction: An equation that has no real number solution

38

Addition Principle of EqualityAddition Principle of Equality

Solve: 1512246355 xx

312241525 xx

33 xAdditive Inverse:

+3 +3+3

0x

solution. the is 0x

Check: 151220463055

15122635 15121215

33