Solving Equations

Vocabulary Solving Equations Translate verbal expressions into algebraic expression and equations and vice versa. Solve equations using the properties of equality. Vocabulary 1) open sentence 2) equation 3) solution

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 equation the variables have been replaced by numbers. Open sentences are neither true nor false until Each replacement that results in a true statement is called a ________ of the open sentence. solution

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

Solving Equations 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 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 c c Example: 4 4 -3 -3

Chapter 5: Solving Equations What will we discuss? What are the parts of an equation What does it mean to solve an equation How do we use inverse operations to solve equations How to solve simple and complex equations 7

What Does it Mean to Solve an Equation? To solve an equation means to find every number that makes the equation true. We do this by adding or subtracting to each side of the equation … but always keep it balanced!

What are the parts of an equation? Let’s first take a look at an equation and identify its parts Variable Constant Coefficient Poll #1: The 3 that is in front of the x is a (a) coefficient, (b) constant, (c) variable, or (d) fraction. Poll #2: The 36 is a (a) coefficient, (b) constant, (c) variable, or (d) fraction. Poll #3: The x is a (a) coefficient, (b) constant, (c) variable, or (d) fraction. Discuss the -1 coefficient on the RHS of the equation. Slide in labels after polls are completed. Discuss equals sign to transition to next slide. 9

So do we just use trial and error to find the right value? No. We can use inverse operations to isolate, or solve for, the variable’s value. Inverse operations? Think about it … The inverse operation of addition is subtraction. And the inverse operation of multiplication is division. Have students brainstorm scenarios in which trial and error would be tedious. 10

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

So how do we solve equations with inverse operations? Let’s take a look at a simple equation Step 1: Now that we have solved the equation, let’s check the solution: - 13 - 13 Answer: 12

So how do we solve equations with inverse operations? Let’s take a look at a simple equation Step 1: Now that we have solved the equation, let’s check the solution: + 5 + 5 Answer: 13

So how do we solve equations with inverse operations? Let’s take a look at a simple equation Step 1: Now that we have solved the equation, let’s check the solution: 25 25 Step 2: Answer: 14

So how do we solve equations with inverse operations? Let’s take a look at a simple equation Now that we have solved the equation, let’s check the solution: Step 1: (16) (16) Step 2: Answer: 15

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

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

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

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

Solving a Proportion Solve the proportion below 12 12 20

Solving a Proportion Solve the proportion below 52 52 21

Checking the Solution to a Proportion Let’s check the solution to the proportion we solved on the last slide 2 22

Using Proportions to Solve Problems You get 46 miles to a gallon of gas. How far can you go on 16 gallons of gas? 23

Multi-step Solutions Let’s take a look at our original equation -12 -12 Step 2: +x +x Is there only one way to solve an equation? Do the steps have to happen in a specific order? Step 3: 4 4 Answer:

Multi-step Solutions (involving distribution) Consider the following equation Step 1: Step 2: +30 +30 Step 3: 6 6 Answer:

Finding Variations of Formulas Solve the formula for r.