Introduction to Algebraic Equations

Section 3.1

Definitions

• Def: A variable expression is a mathematical phrase that uses numbers, variables, and operation symbols. – For example 3x + 2 (x is the variable.)

 

• Def: A term is a group of numbers, or variables, or the product/quotient of numbers and variables (not sum or difference).

On a separate piece of paper…

• Write your name, the period, and the date

• You will answer various questions throughout the lecture on this paper.

• You will turn it in at the end of class.

How many terms are there in the following expressions?

(on paper, not notebook)1.  

2.  

3.  • Note: The number of terms is simply the number of

expressions being added and subtracted.

27 3 10x x

22 ba

7 6139x y z

• Def: If two expressions are equal to each other, you can write this as an equation.

• For example, if x + 3 and 2x – 7 have the same value, then x + 3 = 2x – 7.

 

Back to your notebook:

Which is which? (On separate paper)• Which of the following are expressions? (exp)• Which of the following are equations? (equ)

a) 3x + 5b) 4 = 1+3c) 7x + 87y + 32zd) 6xe) 2x = 8f) 7x + 1 = 9x + 2

Equations always have an equals sign, but expressions do not.

To Solve Algebraic Equations3 + y = 4

- “y” represents an unknown number.

- We can figure out the value of that unknown number.

Solving Equations Algebraically

Sometimes we cannot look at a problem and deduce what the variable represents.

We use “inverse operations” to peel away the layers of information, and discover the value of our unknown.

Inverse Operations (in notebooks)

• The oppositeThey cancel each other out– Multiplication/Division– Addition/Subtraction– Square/square-root– Positives/negatives

For example: x + 3 = 7Addition (x + 3), so we do the inverse (opposite) and subtract.

x + 3 -3 = 7 -3 (Do to both sides)

x = 4

Check: 4 + 3 = 7

Solve the following equations:The Separate Paper Please

4. x + 7 = –2 5. – 3 + x = 8

6. 4x = 20 7. = 1 3x

2x+3=11• 2 multiplied by x, and 3 added to the x

– Opposite of multiplication: division

– Opposite of addition: subtraction

• So, we will need to divide and subtract to solve for x.

2x + 3 = 11 Original Problem

2x + 3 –3 = 11 –3 Subtract 3

2x = 8 One-step left:

2x = 8 Divide by 2

2 2

x = 4 Check: 2(4) + 3 = 11

Solve for the variable:In your notebook, solve the following

equations.

8) 3y – 9 = 63

9) 5a + 6 = 26

10) 2n – 7 = 25

Homework: pg 111-1121-11 odds, 16-32 evens, 39-49 odds