1. 2.1 – Linear Equations in One Variable
Objective – Solve linear equations using properties of equality.
Solve linear equations that can be simplified by combining like terms.
-Solve linear equations involving fractions

2. Why is this important?

3. Linear Equations Linear Equations in one variable –
A linear equation in one variable is an equation that can be written in the form ax + b = c
Where a, b, and c are real numbers and a 0
The Addition and Multiplication Properties of Equality
If a, b, and c, are real numbers, then
a=b and a+c = b – c are equivalent equations
Also a =b and ac = bc are equivalent equations as long as c

4. Solve: Addition property of equality
2 x + 5 = 9
2x = 9 -5
2x = 4
2 2
x = 2
Subtract 5 from both sides
Simplify
Divide both sides by 2
Simplify

5. Check
To see that 2 is the solution, replace x in the original equation with 2.
2 x + 5 = 9
2 (2) + 5 = 9
4 + 5 = 9
9 = 9

6. Give it a try
3 x + 6 = 12
3x + 6 – 6 = 12 –6
3x = 6
3 3
x = 2

7. Check
0.6 = 2 – 3.5c
0.6 = 2 – 3.5 (0.4)
0.6 = 2 – 1.4
0.6 = 0.6

8. Give it a try! 4.5 = 3 + 2.5 x 4.5 – 3 = 3 + 2.5 x – 3 1.5 = 2.5 x
0.6 = x

9. Solve – Combining like terms
-6 x – 1 + 5x = 3
-6x – 1 + 5x = 3
-x – 1 = 3
-x – = 3 + 1
-x = 4
x = -4

10. Give it a try! -2x + 2 – 4x = 20 -6x + 2 = 20 -6x + 2 – 2 = 20 – 2
x = -3

11. Solve: Distributive Property
2 (x – 3) = 5 x – 9
2(x – 3) = 5 x – 9
2x – 6 = 5 x – 9
2x – 6 – 5x = 5x –9 – 5x
-3 x – 6 = - 9
-3 x – =
-3x = -3
x = 1

12. Give it a try!
4 (x – 2) = 6x –10
4x – 8 = 6x – 10
2 = 2 x
1 = x

13. Solve: Adding/Subtracting Fractions

14. Give it a try!

15. Solve: Multiplying Fractions

16. Give it a try!

17. Solve: Decimals
0.3x = 0.27 x – 0.02
100(0.3x +0.1) = 100 (0.27x – 0.02)
100(0.3x) (0.1) = 100(0.27x) – 100(0.02)
30 x = 27 x – 2
30 x – 27 x = -2 –10
3 x = -12
3x = -12
x = - 4

18. Give it a try!
0.2x +0.1 = 0.12x – 0.06
100 (0.2 x + 0.1) = 100 (0.12 x – 0.06)
20 x + 10 = 12 x – 6
20 x – 12 x = -6 – 10
8 x = -16
x = -2

19. Solve: Contradiction 3x + 5 = 3(x+2) 3 x + 5 = 3 (x + 2)
5 = 6
This is a false statement…The original equation has no solution. Its solution set is written either by { } or O. This equation is a contradiction.

20. Give it a try!
5 x – 1 = 5(x+3)
5 x – 1 = 5 x + 15
-1 = 15
{ }

21. This equation is called an identity!
Solve: Identity
6 x – 4 = (x –1)
6x – 4 = 2 + 6x – 6
6 x – 4 = 6 x – 4
{x/ x is a real number}
6 x – – 6x – 4 + 4
6 x = 6 x
6x – 6x = 6 x – 6x
0 = 0
This equation is called an identity!

22. Give it a try! -4(x – 1) = -4x –9 +13 -4x + 4 = -4x + 4
{x / x is a real number}

23. Put it in words! Solving a linear equation in one variable
Step 1: Clear the equation of fractions by multiplying both sides of the equation by the LCD
Step 2: Use distributive property to remove grouping symbols such as parentheses.
Step 3: Combine like terms.
Step 4: Isolate the variable by adding, subtracting, multiplying, dividing (equality properties)
Step 5: Check the solution in the original equation.