Download Sections 2.1, 2.2, 2.3, 2.4 2.1 The Addition Property of Equality 2.2

Sections 2.1, 2.2, 2.3, 2.4
2.1 The Addition Property of Equality
2.2 The Multiplication Property of Equality
2.3 More on Solving Linear Equations
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
expression and an equation.
Linear Equations in one variable: 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 real
number, 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 subtraction 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,
represent real numbers, then if A = B, then AC =BC
For example: 1 x = 8
(C ≠ 0) ,
4
4. The division property of equality: If A, B, and C,
real numbers, and A = B, then A = B
C
For example:
(C ≠ 0)
represent
C
3 x = 18
***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
11.
k
= −3
8
12.
5
4
− 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 fraction, you can
multiply by the least common multiple to “Clear the fractions”.
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.
− 5(3w − 3) + (1 + 16 w ) = 0
15. -5p + 4 = 19
16. 2q + 3 = 4q – 9
17. 7( p − 2 ) + p = 2 p + 4
18.
2 − 3(2 + 6 z ) = 4( z + 1) + 18
19.
1
(x + 3) − 2 (x + 1) = −2
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 integers such that 6 times the lesser
added to the greater gives a sum of 86.