Solving Linear Equations Finding the Unknown Solving Linear Equations

Definitions Equation: A mathematical sentence with an equals sign, which states that 2 expressions are equal: 12 – 3 = 9 18 + 6 = 24

Balance these equations Balance these equations so that both sides are equal Equals Left Hand Side Right Hand Side 12+ 4 + 6 = 21 9 + 7 + 2 = 6 + 2 3 – 13 + 5 = 3 + 2

Activity: Balance the Scales

Scales in Balance An Equation is like a balance scale. Everything must be equal on both sides If we change one side of the equation or balance (by adding or subtracting) we must also do the same to the other side. If we don't the scales become unbalanced and are no longer equal!

Finding an Unknown How can we solve for X easier? Find the value of x for each of the following 10x + 20 = 40 5x + 5 = 20 6x + 30 = 36 2x + 12 How can we solve for X easier? Use Inverse Operations!

Inverse Operations Operation Inverse Operation Addition Subtraction Inverse Operation : An operation that reverses the effect of another operation. Some simple inverse operation are below. Operation Inverse Operation Addition Subtraction Division Multiplication

Solving equations with an unknown on one side One Step Equations Solving equations with an unknown on one side

(The one that will undo what is being done to the variable) ONE STEP EQUATIONS To solve one step equations, you need to ask three questions about the equation: 1 What is the variable? 2 What operation is performed on the variable? 3 What is the inverse operation? (The one that will undo what is being done to the variable)

ONE STEP EQUATIONS x + 4 = 12 - 4 - 4 x = 8 Example 1 Solve x + 4 = 12 What is the variable? The variable is x. What operation is being performed on the variable? Addition. What is the inverse operation (the one that will undo what is being done to the variable)? Subtraction. Using the subtraction property of equality, subtract 4 from both sides of the equation. The subtraction property of equality tells us to subtract the same thing on both sides to keep the equation equal. x + 4 = 12 - 4 - 4 x = 8

a = 24 4 = 6 4a = 24 Example 1 Division This means “4 multiplied by a” What is the inverse operation? = 6 Division

What is the inverse operation? Example 2 b = 3 5 This means “b divided by 5” b = 3 x 5 What is the inverse operation? = 15 Multiplication

Solving equations with an unknown on one side Two Step Equations Solving equations with an unknown on one side

8x + 5 = 61 - 5 - 5 = 56 8x Let’s try a problem. The Problem: This problem has addition, so we need to subtract first. = 56 8x Remember: Whatever we do on one side, we have to do on the other.

Now, Step 2. 8x = 56 This problem has multiplication, so we need to divide now. 8 8 7 x = Remember: Whatever we do on one side, we have to do on the other.

Let’s try another problem. + 8 + 8 This problem has subtraction, so we need to add first. 3a 12 = Remember: Whatever we do on one side, we have to do on the other.

This problem has Multiplication, so we need to divide now . Now, Step 2. 3a = 12 3 3 This problem has Multiplication, so we need to divide now . a 4 = Remember: Whatever we do on one side, we have to do on the other.

Example 3 3a + 8 = 35 Let’s make this question into a simpler expression. We know how to solve those! 3a = 35 - 8 This is the only bit that is different. We can use inverse operations to get rid of it. This says “add 8” 3a = 27 What is the inverse operation? a = 27 3 Subtraction Look! Now your question looks simple to answer! = 9

Solving word problems using linear equations

Solving Word Problems How to solve worded problems: Identify the unknown quantity and use a Pronumeral to represent it. Search for keywords that indicate the steps needed for the solution. Create a linear equation from the information provided in the question. Solve the equation. Interpret the result and write the worded answer.

Addition Word Problem Linear Equation What is the sum of 8 and y? Express the number (x) of apples increased by two x + 2 Express the total weight of Alphie the dog (x) and Cyrus the cat (y) x + y Key words for addition + : increased by; more than; combined together; total of; sum; added to

Subtraction Word Problem Linear Equation What is four less than y What is nine less than a number (y) y - 9 What if the number (x) of children was reduced by 6? x - 6 What is the difference of my weight (x) and your weight (y) x - y Key words for Subtraction - : less than, fewer than, reduced by, decreased by, difference of

Multiplication Word Problem Linear Equation What is y multiplied by 13 13y or 13 * y Three runners averaged "y" minutes. Express their total running time: 3y I drive my car at 55 miles per hour. How far will I go in "x" hours? 55x Key words for multiplication  * x or integers next to each other (5y, xy) : of, times, multiplied by

Division Word Problem Linear Equation What is the quotient of y and 3 y/3 or y ÷ 3 Three students rent an apartment for $ "x" /month. What will each have to pay? x/3 or x ÷ 3 "y" items cost a total of  $25.00. Express their average cost: 25/y or 25 ÷ y Key words for division  ÷ / per, a; out of; ratio of, quotient of; percent (divide by 100)

Complex question example Al's father is 45. He is 15 years older than twice Al's age. How old is Al? We can begin by assigning a Variable/Pronumeral to what we're asked to find. Here this is Al's age, so let Al's age = x. We also know from the information given in the problem that 45 is 15 more than twice Al's age. How can we translate this from words into mathematical symbols? What is twice Al's age? Well, Al's age is x, so twice Al's age is 2x, and 15 more than twice Al's age is 15 + 2x. That equals 45, right? Now we have an equation in terms of one variable that we can solve for x: 45 = 15 + 2x.

The Solution – Finding the value of x Original Problem: 45 = 15 + 2x First step is to get rid of the number 15. To do this we need to subtract 15 from both sides of the equation. 45 – 15 = 15 – 15 + 2x We now have: 30 = 2x The Second Step is to get the variable/Pronumeral by itself (in this case a single x). To do this we will need to divide 2x by 2, which means we have to divide both sides of the equation by 2. 15 = x or x = 15 Since x is Al's age and x = 15, this means that Al is 15 years old.

Practice these word problems on your own: Real World Situations Practice these word problems on your own: There are 26 students in Ms. Bean's class. The number of boys is equal to seven fewer than twice the number of girls. How many boys and how many girls are in the class? You are ordering tulip bulbs from a flowering catalog. The cost is .75 cents per bulb. You have $14 to spend. If the shipping cost is $3 for any size order, determine the number of bulbs you can order.

Ms Bean’s Class x + 2x – 7 = 26 x is # of girls; 2x-7 is # of boys +7 +7 Addition Property of Equality 3x = 33 Division Property of Equality 3 3 x = 11 So there are 11 girls and 15 boys.

Ordering Tulips .75b + 3 = 14 - 3 - 3 Addition Property of Equality .75b = 11 Division Property of Equality .75 .75 b = 14.667 So you can purchase about 14 bulbs. If you purchase 15 bulbs you will go over your $14 budget.