Solving Linear Equations

1. Finding the Unknown Solving Linear Equations

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

3. Balance these equations Balance these equations so that both sides are equal Equals

4. Activity: Balance the Scales

5. 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!

6. Finding an Unknown Find the value of x for each of the following How can we solve for X easier? Use Inverse Operations!

7. Inverse Operations Inverse Operation : An operation that reverses the effect of another operation. Some simple inverse operation are below.

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

9. ONE STEP EQUATIONS To solve one step equations, you need to ask three questions about the equation: What is the variable? 1 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)

10. ONE STEP EQUATIONS 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

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

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

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

14. Let’s try a problem. 8x + 5 = 61 The Problem: - 5 - 5 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.

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

16. Let’s try another problem. 3a - 8 = 4 + 8 + 8 This problem has subtraction, so we need to addfirst. 3a 12 = Remember: Whatever we do on one side, we have to do on the other.

17. 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.

18. a = 27 3 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? Subtraction Look! Now your question looks simple to answer! = 9

19. Solving word problems using linear equations

20. 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.

21. Addition Key words for addition + :increased by; more than; combined together; total of; sum; added to

22. Subtraction Key words for Subtraction - :less than, fewer than, reduced by, decreased by, difference of

23. Multiplication Key words for multiplication  * x or integers next to each other (5y, xy) :of, times, multiplied by

24. Division Key words for division  ÷ /per, a; out of; ratio of, quotient of; percent (divide by 100)

25. 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.

26. 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.

27. 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.

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

29. Ordering Tulips .75b + 3 = 14 - 3 - 3 Addition Property of Equality .75b = 11Division 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.