1. 2
Equations, Inequalities, and Applications

2. 2.1 The Addition Property of Equality
Objectives
1. Identify linear equations. 2. Use the addition property of equality. 3. Simplify, and then use the addition property of equality.

3. Identify Linear Equations
Linear Equation in One Variable
A linear equation in one variable can be written in the
form
Ax + B = C
where A, B, and C are real numbers, with A ≠ 0.
Some examples of linear and nonlinear equations follow.
4x + 9 = 0, 2x – 3 = 5, and x = Linear
x2 + 2x = 5, = 6, and |2x + 6| = 0 Nonlinear

4. Identify Linear Equations
A solution of an equation is a number that makes the
equation true when it replaces the variable. Equations
that have exactly the same solution sets are equivalent
equations.
A linear equation is solved by using a series of steps
to produce a simpler equivalent equation of the form
x = a number or a number = x.

5. Use the Addition Property of Equality
If A, B, and C are real numbers, then the equations
A = B and A + C = B + C
are equivalent equations.
In words, we can add the same number to each side
of an equation without changing the solution set.

6. Use the Addition Property of Equality
Note
Equations can be thought of in
terms of a balance. Thus, adding
the same quantity to each side
does not affect the balance.

7. Use the Addition Property of Equality
Example 1 Solve the equation.
Our goal is to get an equivalent equation of the form x = a number.
x – 23 = 8
x – =
x = 31
Check: 31 – 23 = 8 

8. Use the Addition Property of Equality
Example 2 Solve the equation.
y – 2.7 = –4.1
y – = –
y = – 1.4
Check: –1.4 – 2.7 = –4.1 

9. Use the Addition Property of Equality
The same number may be subtracted from each side of
an equation without changing the solution.
If a is a number and –x = a, then x = –a.

10. Use the Addition Property of Equality
Example 3 Solve the equation.
Our goal is to get an equivalent equation of the form x = a number.
–12 = z + 5
–12 – 5 = z + 5 – 5
–17 = z
Check: –12 = –17 + 5 

11. Subtracting a Variable Term
Example 4 Solve the equation.
4a + 8 = 3a
4a – 4a + 8 = 3a – 4a
8 = –a
–8 = a
Check: 4(–8) + 8 = 3(–8) ?
–24 = –24 

12. Simplify and Use the Addition Property of Equality
Example 7
Solve.
Check:
5((2 · –36) –3) – (11(–36) + 1) = 20
5(–72 –3) – (– ) = 20
5(2b – 3) – (11b + 1) = 20
5(–75) – (–395) = 20
10b – 15 – 11b – 1 = 20
– = 20
–b – 16 = 20
20 = 20
–b – = 
–b = 36
b = –36