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Home , Equality (mathematics)

Sections 2.1, 2.2, 2.3, 2.4 2.1 The Property of Equality 2.2 The Multiplication Property of Equality 2.3 More on Solving Linear 2.4 An Introduction to Applications of Linear Equations

The Addition Property of Equality

In chapter 1, we saw that there was a difference between an and an .

Linear Equations in one : Ax + B = C

To solve a linear equation means to find the value of the variable that makes the equation true.

Some basic properties that allow you to solve a linear equation First, think of any equation as a scale balance. Almost any mathematical operation is legitimate as long as you perform the same operation on BOTH SIDES of the equal sign.

1. The addition property of equality: If A = B, and if C is any , then A + C = B + C For example: if x = 5, then x + 2 = 5 + 2 But more useful: if x -2 = 3, then

2. The property of equality: if A = B and if C is any real number, then A – C = B – C For example: if x +2 = 7, then

3. The multiplication property of equality: If A, B, and C, C ≠ )0( , represent real numbers, then if A = B, then AC =BC 1 For example: x = 8 4

4. The division property of equality: If A, B, and C, C ≠ )0( represent A B real numbers, and A = B, then = C C For example: x = 183

***The important thing to remember is that when you solve an equation is that you are trying to get all the variables on one side and all the numbers on the other. It doesn’t matter if the variable is left on the left or right side of the equation***

Examples: Addition/Subtraction Property of Equality

1. x – 8 = 9

2. x + 45 = 24

3. 8 + k = -4

4. 3 = z + 17

5. 3x = 2x + 17

6. 8t +5 = 7t

Examples of Multiplication/Division Property of Equality 7. 8x = 24

8. -8x = -64

9. 5x = 0

10. –t = 14

k 11. −= 3 8

5 4 12. d =− 6 9

13. 9p – 13p= 24

Using the rules together: 1. Simplify the expression. What I mean by that is distribute and remove any parentheses in the problem. Another thing to be aware of: If you have a coefficient that is a , you can multiply by the least common multiple to “Clear the ”. We will do an example that involves this. 2. Add/subtract any constants to remove them from one side of the equation. Remember to perform the operation on both sides of the equation 3. Add/subtract any variables so that they are on the other side of the equation from the constants. Again, remember to perform the operation on both sides of the equation. 4. Multiply or divide both sides of the equation by a constant that makes the coefficient on the variable a 1. 5. When you make it to 1x = #, you have your solution! Examples:

Examples of using the rules together

14. ()()w w =++−− 0161335

15. -5p + 4 = 19

16. 2q + 3 = 4q – 9

17. () ppp +=+− 4227

18. ()()zz ++=+− 18146232

1 2 19. ()3 ()xx −=+−+ 21 4 3

Section 2.4 is an introduction to Applications of Linear Equations Application problems are what you may think of as “word problems”. First we will discuss a good general approach to setting up application problems as equations and then we will work on solving them.

6-step method to solving application problems

Examples: 1. If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the number.

2. In the 2006 Winter Olympics in Torino, Italy, Canada won 5 more medals than Norway. The two countries won a total of 43 medals. How many medals did each country win?

3. A piece of pipe is 50 in. long. It is cut into three pieces. The longest piece is 10 in. longer than the middle sized piece and the shortest piece measures 5 in. less than the middle sized piece. Find the length of the three pieces.

4. Find two consecutive even such that 6 times the lesser added to the greater gives a sum of 86.