1. Solving Linear Equations
1.5 through 1.7

2. Equations Is 2 + 4 = 6 a true equation? Is 3 – 5 = 10 a true equation?
Is 2x + 1 = 7 a true equation?
This equation is a conditional equation, since it depends on x.

3. Solving Linear Equations:
The graph of any linear equation is a straight line.
Solving Linear Equations:
If there are fractions, multiply everything by the LCD.
Get rid of ( )’s and combine like terms.
Isolate the variable (get it by itself on one side of equation).
Check your answer!

4. Example 1 Determine whether 3 is a solution of 2x + 4 = 10
Solution To find out, check the proposed solution, x = 3:
2x + 4 = 10 This is the original equation.
2(3) + 4 = 10 Substitute 3 for x.
6 + 4 = Multiply.
10 = 10 Add.
Since 10 = 10, 3 is a solution, or root, of the equation.

5. Example 3 Solve 3(x - 2) = 20. 3(x - 2) = 20 3x – 6 = 20 3x = 26 x =
Solution:
3(x - 2) = 20
Original equation.
3x – 6 = 20
Use the distributive property.
3x = 26
Add 6 to both sides.
x =
Divide both sides by 3.

6. Example 5 Solve 10(x - 3) = 9(x - 2) + 12 10x - 30 = 9x - 18 + 12
Solution:
This is the given equation.
Multiply both sides by 6 (LCD).
Use distributive property.
10(x - 3) = 9(x - 2) + 12
10x - 30 = 9x
Use distributive property again.
10x – 30 = 9x - 6
Combine like terms.
x = 24
Isolate variable.

7. Solving Linear Equations:
The graph of any linear equation is a straight line.
Solving Linear Equations:
If there are fractions, multiply everything by the LCD.
Get rid of ( )’s and combine like terms.
Isolate the variable (get it by itself on one side of equation).
Check your answer!

8. Types of Equations
Conditional: An equation that is true for at least one value of “x” (it’s sometimes true).
Identity: An equation that is true for all values of “x” (it’s always true).
Contradiction: An equation that has no solution (it’s never true).

9. Example 7 Solve 2(x +1) – x = 3(1 + x) – (2x + 1):
Original equation.
2x + 2 – x = x – 2x – 1
Use the distributive property.
x + 2 = x + 2
Combine like terms.
Last equation is always true for any value of x
it is an identity!

10. Example 8 Solve Since –2 = 5 is never true, there is no solution
Original equation.
Multiply everything by 6 (LCD).
Simplify fractions.
Use distributive property.
26x – 2 = 26x + 5
Combine like terms.
-2 = 5
Subtract 26x from both sides.
Since –2 = 5 is never true, there is no solution
it is a contradiction!

11. Time Trial This equation is a contradiction.
Determine whether the equation 3(x - 1) = 2(x + 3) + x
is an identity, a conditional equation, or a contradiction.
Solution To find out, solve the equation.
3(x – 1) = 2(x + 3) + x
3x – 3 = 2x x
3x – 3 = 3x + 6
-3 ≠ 6
This equation is a contradiction.

12. Formulas
A formula is an equation involving two or more variables.
D = RT
A = LW
P = 2L + 2W

13. Example 10 Solve for t in the following formula: A = p + prt.
Solution: To solve for t means to isolate it on on side.
A = p + prt
Original equation.
A – p = prt
To isolate t, subtract p from both sides.
Divide both sides by pr.
Write t on left-hand side.