1. Solving Equations

3. Solving Equations A mathematical sentence (expression) containing one or more variables is called an open sentence . A mathematical sentence stating that two mathematical expressions are equal is called an _________. equation Open sentences are neither true nor false until the variables have been replaced by numbers. Each replacement that results in a true statement is called a ________ of the open sentence. solution

4. To solve equations, we can use properties of equality. Some of these equivalence relations are listed in the following table. Reflexive Symmetric Transitive Substitution For any real number a , a = a , For all real numbers a and b , If a = b, then For all real numbers a , b , and c . If a = b, and b = c, then If a = b, then a may be replaced by b and b may be replaced by a. b = a a = c – 5 + y = – 5 + y If 3 = 5x – 6, then 5x – 6 = 3 If 2x + 1 = 7 and 7 = 5x – 8 then, 2x + 1 = 5x – 8 If (4 + 5) m = 18 then 9m = 18 Solving Equations Properties of Equality Property Symbol Example

5. Sometimes an equation can be solved by adding the same number to each side or by subtracting the same number from each side or by multiplying or dividing each side by the same number. Addition and Subtraction Properties of Equality For any real numbers a , b , and c , if a = b, then a = b + c + c a = b - c - c Example: If x – 4 = 5, then x – 4 = 5 + 4 + 4 If n + 3 = –11, then n + 3 = –11 – 3 – 3 Solving Equations

6. Sometimes an equation can be solved by adding the same number to each side or by subtracting the same number from each side or by multiplying or dividing each side by the same number. Multiplication and Division Properties of Equality For any real numbers a , b , and c , if a = b, then a = b · c · c a = b Example: 4 4 Solving Equations c c - 3 -3

11. Solving 1 Step Equations How much does the suitcase weigh in terms of blocks? B=Blocks S=Suitcase Equation: 6B + S = 9B -6B -6B 3B S = What is the weight of the suitcase if each block has a weight of 2lbs. ? S = 3 (2) = 6 lbs.

16. 1 Step Equation X + 11 = 9 X - 37 = 52 3X = 72 -11 -11 X = -2 3 3 X = 24 1 1 20 + h = 41 17 - s = 27 This is the same as -1S=10

17. 1 Step Equations Continued… 6X = 42 P = 6 6 1 X=7 or x = 7 1 1 Cross Multiply Multiply by the reciprocal of 2/5 2 5 3 4

18. Multi Step Equations Solve: 8m – 10 = 36 8m – 10 = 36 8m = 46 8 8 m = + 10 + 10

19. Multi Step Equations 5x  2 = x + 4 Solve: 5x  2 = x + 4 Notice that there are variables on both sides 5x = x + 6 Get rid of the -2 on the left side Simplify 5x = x + 6 Get rid of the x on the right side 4x = 6 Get rid of the coefficient of x 4 4 x = Simplify Simplify + 2 + 2 – x – x