1. Lesson 4: Distributive Property, Solutions of Equations, Change Sides – Change Signs

2. Use the distributive property to solve the expression 3(4 + 5) We say that we distributed the multiplication over the addition.

3. Example: Expand 4a b - 3ba b a a -2

4. Answer: a b a

5. Example: Expand a b a bc - 3a c c b

6. Answer: a b - 3a c cb

7. Addition Rule for Equations: The same quantity can be added to both sides of an equation without changing the solution set of the equation = = = 9 Still True

8. Multiplication and Division Rule for Equations: Every term on both side of an equation can be multiplied (or divided) by the same nonzero quantity without changing the solution set of the equation = 7 2(4 + 3) = 7(2) 14 = 14 Still True

9. We remember that we always use the addition rule before we use the multiplication/division rule. This is because the solution of an equation undoes a normal order-of-operations problem.

10. We remember from Algebra 1 that the five steps for solving simple equations in one variable are:
Eliminate parentheses
Add like terms on both sides
Eliminate the variable on one side or the other
Eliminate the constant term on the side with the variable
Eliminate the coefficient of the variable

11. Example: Solve 12 – (2x + 5) = -2 + (x – 3)

12. Answer: 4 = x

13. Example: Solve 3 5 - 5 x = - - 1 + x 6 3 2

14. Answer: ½ = x

15. It is important to understand why we do things in algebra, but it is also important not be let the emphasis on understanding interfere with our ability to do. The use of the addition rule for equation is a case in point. We can use this rule to eliminate a term from one side of an equation by adding the opposite of the term to both sides of the equation.

16. For example, if we wish to solve for y in the equation y + 2x = 4 We add -2x to both sides of the equation. y = x

17. We were able to eliminate the 2x term from the left hand side of the equation, but we we did, the same term appeared on the right hand side of the equation with its sign changed. This happens every time we use the addition rule. The term will disappear on one side of the equation and reappear on the other side with its sign changed. This leads to the adage change sides – change signs.

18. Mathematicians in the late 1800’s called this process transposition.

19. Example: Use the rule “change sides – change signs” to solve for x: x – 2 = 7

20. Answer: x = 9

21. Example: Use transposition to solve for p: p – 3x + 4 = 7y

22. Answer: p = 7y + 3x – 4

23. Example: Solve for y: 3y – 2x + 5 = 0

24. Answer: y = 2/3x – 5/3

25. HW: Lesson 4 #1-30