1. Introduction to Algebra
Grade 7
Mr. Pontrella

2. Solving Equations Using Inverse Operations
An equation is a mathematical sentence that contains an = sign.
555 = A-75 A-75 = 555 are examples of an equation.
In the examples above “A” is a variable which represents an unknown number.
One way to solve for A is to complete the Inverse (opposite) Operation (the opposite of subtraction is addition)
= A or A = , so A = 630

3. Addition and Subtraction
Inverse Operations
Addition and Subtraction
1. T - 5 = 11
= T
16 = 16
T = 11
T =
T = 5
Memorize this type of problem so you can solve using the same operation
Find the example
that does not
use the inverse
Operation.
B = 15
B =
B = 6
3. B + 6 = 15
B =
B = 9

4. Multiplication and Division
Inverse Operations
Multiplication and Division
5. 6 x R = 48
= R
8 = R
S = 9
= S
8 = S
Find the example
that does not
use the inverse
Operation.
Memorize this type of problem so you can solve using the same operation
6. W x 8 = 48
= w
6 = w
8. M = 8
8 x 9 = M
72 = M

5. Property of Equality
Equations are like a Balance scale. If weight is added or subtracted to one side it will make the scale tip or not be balanced. In order to keep the scale balanced you must add or subtract equal amounts to both sides.
This idea will help us solve equations with variables.

6. Property of Equality for Addition and Subtraction
Add 244 to both sides
F =
Solve
Addition Property of Equality If you add the same number to each side of an equation, then the 2 sides remain the same.
F = 364
Subtract 3.6 from both sides
X = 12.4
X =
Solve
Subtraction Property of Equality If you subtract the same number from each side of an equation, then the 2 sides remain the same.
X = 8.8

7. Property of Equality for Multiplication and Division
Remember a variable next to a number indicates multiplication
434 = 2s
Divide both sides by 2
Solve
434 = 2s
Division Property of Equality If each side of an equation is divided by the same nonzero number, then the 2 sides remain equal.
S = 217
Multiply both sides by 8
n = 96
n x 8 = 96 x 8
Solve
Multiplication Property of Equality If each side of an equation is multiplied by the same number, then the 2 sides remain equal.
n = 768

8. Writing Algebraic Expressions
An expression is one part of an equation
Examples a
7 - x d
Numerical expressions are often written in sentence form.
Five more hits than the Yankees Ten fewer points than the Knicks
Yankees Knicks - 10 or
Y K - 10
The key words often indicate what to do
more means add
fewer means subtract

9. Key words to Look For Multiplication Addition Times Plus Product Sum
multiplied
Addition
Plus
Sum
More than
Increased by
Total
Subtraction
Minus
Difference
Less than
Subtract
Decreased by
Division
Divided
quotient

10. Solving Two-Step Equations
3x = 18 3x + 9 = 18
How are these equations different?
When one side of an equation has 2 or more operations we need more than one step to solve.
3x + 9 = 18
First subtract 9 from both sides.
3x =
Simplify
3x = 9
Then divide each side by 3
3x = 9
So x = 3

11. Solving Two-Step Equations
B = 1.3 9
First add 0.8 to both sides
B = 9
Simplify
B = 2.1 9
Then multiply both sides by 9
So B = 18.9
B x 9 = 2.1 x 9 9

12. Integers -5 is the opposite of 5
An integer is a whole number that can be either greater than 0, called positive,
or less than 0, called negative. Zero is neither positive nor negative.
Two integers that are the same distance from zero in opposite directions are called
opposites.
-5 is the opposite of 5
Every integer on the number line has an absolute value,
which is its distance from zero. The brackets indicate the absolute value.

14. Using a Number Line to Add or Subtract Integers
Add a positive integer by moving to the right on the number line
Add a negative integer by moving to the left on the number line 2 6
2 + 6 = ?
8 + (- 3 ) = ? 8 -3

15. Subtract an integer by adding its opposite
3 - 7 = ? -7 3
3 - (-7) = ? 7 3

16. Another way to remember the rule for subtraction is Keep, change, change!
= ?
Change the
sign of the
second number
Keep the first number
Change the sign + -7 3
4 - 9 = ? 4 +
(-9)
Keep the 4
Change the sign of the 9
Change the sign

17. Multiplication and Division
of Integers
To multiply or divide signed integers, always multiply or divide the absolute values
and use these rules to determine the sign of the answer:
If the signs are the same the product or quotient is Positive
6 x 3 = 18 (-6 ) x (-3) = 18
If the signs are different the product or quotient is Negative
(-6 ) x 3 = x (-3) = -18

18. Using Counters to Solve Integer Problems
A positive and a negative counter equal 0
To add we put counters in the box
To add 4 + (-3) we start with 4 positive counters
We put 3 negative counters in
The positive and negative pairs cancel each other out to make 0
We are left with 1 positive counter

19. Subtracting Integers With Counters
To subtract we take counters out of the box
4 - (-5)
Since there are no negative counters we must add negatives to the box. If we add 0 to the box we do not change the value
If we take out the 5 negatives we will be left with 9 positives
4 - (-5) = 9

20. Using Counters to Multiply Integers
In the sentences 3 x 2 we will place 3 groups of 2 in the box

21. Multiplying a Positive Integer by a Negative Integer
In the problem 3 x (-2) we are putting in 3 groups of 2 negatives in the box
The box shows that 3 x (-2) = - 6

22. Multiplying With a Negative Integer
When multiplying by a negative integer we are taking out groups
Since the box is empty we must add 0 pairs until we have enough to take out 2 groups of 3
In the problem (-2 ) x 3 we are taking out 2 groups of 3
We can now take out 2 groups of 3
We are left with 6 negatives
(-2) x 3 = -6

23. Solving Inequalities How are these number sentences different?
3x + 9 = 18 3x + 9 > 18
When an equation has > or < sin it is called an inequality
We solve inequalities in the same way as equations
3x + 9 > 18
First subtract 9 from both sides.
3x >
Simplify
3x > 9
Then divide each side by 3
3x > 9
So x > 3

24. Graphing Inequalities
We can graph the solution to an inequality on a number line.
2a - 5 > 9
First, solve as you would an equation
2a > 9 + 5
Add 5 to both sides
2a > 9
Divide both sides by 2
a > 2.5
Since a is greater than 2.5 it will include all numbers greater than 2.5 but not 2.5
We can draw a circle at 2.5 to show this.

25. Coordinate Plane (-3,1) coordinate plane
The plane determined by a horizontal number line, called the x-axis,..
and a vertical number line, called the y-axis,
Each point in the coordinate plane can be specified by an ordered pair of numbers
intersecting at a point called the origin
(-3,1)

26. (1,3) Name each Point. (-5,2) (-2,4) (3,-4) (5,-6) (5,-4)
Here's one way geometry is used in the real world. A team of archaeologists is studying the ruins of Lignite, a small mining town from the 1800's. They plot points on a coordinate plane to show exactly where each artifact is found.
(1,3)
Name each
Point.
(-5,2) (-2,4)
(3,-4) (5,-6)
(5,-4)