1. Copyright 2013, 2009, 2005, 2002 Pearson, Education, Inc.

2. Linear Equations in One Variable
2.1
Linear Equations in One Variable

3. Linear Equations
Algebraic equation is a statement that two expressions have equal value.
Solving algebraic equations involves finding values for a variable that make the equation true.
Linear equation in one variable can be written in the form ax + b = c, a  0.
Equivalent equations are equations with the same solutions in the form of
variable = number, or
number = variable.

4. 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 and a ≠ 0.

5. Addition Property of Equality
a = b and a + c = b + c are equivalent equations
Multiplication Property of Equality
a = b and ac = bc are equivalent equations

6. Example 8 + z = – 8 8 + (–8) + z = –8 + –8 Add –8 to each side
z = –16 Simplify both sides

7. Example
Recall that multiplying by a number is equivalent to dividing by its reciprocal
3z – 1 = 26
3z – = (Add 1 to both sides)
3z = 27 (Simplify both sides)
(Divide both sides by 3)
z = (Simplify both sides)

8. Example
Solve: 6x + 8 – 5x = 8 – 3
Objective A

9. Example Solve: 4p – 11 – p = 2 + 2p – 20
3p – 11 = 2p – Simplify both sides.
3p + (– 2p) – 11 = 2p + (– 2p) – 18 Add –2p to both sides.
p – 11 = – Simplify both sides.
p – = – Add 11 to both sides.
p = – Simplify both sides.

10. Example
Solve: 3(3x – 5) = 10x
Objective A

11. Example Solve: 5(3 + z) – (8z + 9) = – 4z
15 + 5z – 8z – 9 = – 4z Use distributive property.
6 – 3z = – 4z Simplify left side.
6 – 3z + 4z = – 4z + 4z Add 4z to both sides.
6 + z = Simplify both sides.
6 + (–6) + z = 0 + (–6) Add –6 to both sides.
z = – Simplify both sides.

12. Example 12x + 30 + 8x – 6 = 10 20x + 24 = 10 (Simplify left side)
20x (– 24) = 10 + (– 24) (Add –24 to both sides)
20x = – (Simplify both sides)
(Divide both sides by 20)
(Simplify both sides)

13. Solving a Linear Equation
Clear the equation of fractions by multiplying both sides of the equation by the LCD of all denominators in the equation.
Use the distributive property to remove grouping symbols such as parentheses.
Combine like terms on each side of the equation.
Use the addition property of equality to rewrite the equation as an equivalent equation with variable terms on one side and numbers on the other side.
Use the multiplication property of equality to isolate the variable.
Check the proposed solution in the original equation.

14. Example (Multiply both sides by 5) (Simplify) (Add –3y to both sides)
(Simplify; add –30 to both sides)
(Simplify; divide both sides by 7)
(Simplify both sides)

15. Example
5x – 5 = 2(x + 1) + 3x – 7
5x – 5 = 2x x – 7 (Use distributive property)
5x – 5 = 5x – (Simplify the right side)
Both sides of the equation are identical. Since this equation will be true for every x that is substituted into the equation, the solution is “all real numbers.” The equation is called an identity.

16. Example 3x – 7 = 3(x + 1) 3x – 7 = 3x + 3 (Use distributive property)
3x + (– 3x) – 7 = 3x + (– 3x) (Add –3x to both sides)
–7 = (Simplify both sides)
Since no value for the variable x can be substituted into this equation that will make this a true statement, there is “no solution.” The equation is a contradiction.