Algebra
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Equations with Variables
PATTERNS & VARIABLES
Pattern Rules
Input & Output Machines
Example 1) Find the Input to Output Pattern Rule:
2
?
?
13
Draw out the input/output machine
Put your first input term on the input side and your first output term on the output side
2
x6
?
12
13
Input
Output
2
13
4
25
6
37
8
49
Step 1:
Step 2: Use trial and error to solve the first operation (x6)
2
x6
+1
12
13
Step 3: Use trial and error to solve the second operation to reach the output (13)
Step 4: Then try with other inputs (4, 6, 8) to see if each one equates the correct outputs (25, 37, 49)
Step 5: Write the solution as the input to output pattern rule:
"Multiply the input by 6, then add 1"
6n+1
Practice
Practice PDFs available for download:
Variables I
Variables II
Example 1) Find the input to output pattern rule using variables
Input
Output
2
13
4
25
6
37
8
49
2
?
?
13
Step 1) Use trial and error method to solve the 2 step input to output machine
x6
+1
12
13
Step 2) Now translate the machine into an algebraic expression:
"Multiply the input by 6, then add 1"
6n+1
"n" represents any input value
Step 3) Input the other input terms to check if they produce the correct outputs
Input
Output
2
13
4
25
6
37
8
49
6n+1
6(2)+1
6x2=12 +1 = 13
6n+1
6(4)+1
6x4=12 +1 = 25
6n+1
6(6)+1
6x6=12 +1 = 37
6n+1
6(8)+1
6x8=12 +1 = 49
Practice
Practice PDFs available for download:
EQUATIONS & VARIABLES
Equations with Variables
Equivalent Equations I
Example 1) Is this scale balanced
3 x 7
10 + 10 +1
Step 1) Solve each side of the scale
21
3 x 7
21
10 + 10 +1
21
21
=
Step 2)
Determine whether each product are equal in value
If they are equal, the scale is balanced
If they are not equal, the scale is not balanced
Commutative Property: When add or multiply two numbers together, the order does not effect their sum or product
Equivalent Equations II
Example 1) Find a equivalent equation for the following equation:
Rule to preserve equality:
"Whatever you do to one side, do to the other."
~Mr. Ram's Mom
3n = 12
n = 4
3(4) = 12
12 = 12
Both sides equal each other, therefore equality is preserved
3n = 12
n = 4
3n -1 = 12 -1
3(4) -1 = 12 -1
12 -1 = 12 -1
11 = 11
After (-1) on each side of the equation, both sides are still equal therefore equality is preserved
Practice
Practice PDFs available for download:
Solving Equations with Variable Isolation
Example 1) Solve for x
3x + 5 = 20
Step 1) Isolate x by subtracting 5 from both sides.
​
3x + 5 = 20
- 5
- 5
3x = 15
Step 2) To isolate x , divide both sides by 3:
3x = 15
3
3
x = 5
DISTRIBUTIVE PROPERTY
Distributive Property
Example 1) Solve for x
Step 1) Use the distributive property to expand.
​
2(3x + 5) = 30
3(3x + 5) = 20
2*3x = 6x​
2*5 = 10
6x + 10 = 34
Step 2) Isolate the variable to solve for x.​
6x + 10 = 34
- 10
- 10
6x = 24
6
6
x = 4
FUNCTION NOTATION
Function Notation
POLYNOMIALS
Adding Polynomials
Multiplying Polynomials
Subtracting Polynomials
Dividing Polynomials
Expanding Polynomials
Expanding Binomials I
Expanding Binomials II
How to Factor Trinomials
x - 2x - 8
2
(x + a)(x + b)
_
_
sum of -2
a + b = -2
produce -8
a x b = -2
When we factor trinomials, a and b, must add up to make -2, and multiply together to produce -8.
​
So, you must find values for a & b that fit!
Example 1) Factor this trinomial:
x - 2x - 8
2
(x + a)(x + b)
_
_
Step 1) We must find values of a & b that add up to make -2 and multiply together to produce -8.
​
Lets then begin by listing factors of -8 and see which ones can also add up to -2.
Factors of -8:
1 x (-8) = -8
(-1) x 8 = -8
​
1 + (-8) = -7
(-1) + (8) = 7
The pink ones produce (-8) but none add up to -2
2 x (-4) = -8
(-2) x 4 = -8
​
2 + (-4) = -2
(-2) + (4) = 2
The pink ones produce (-8) and the green one add up to -2
so our a & b values are 2 and (-4)
Step 2) Write it in the correct form:
(x + 2)(x - 4)
Step 3) Expand to check (use the distributive property):
(x + 2)(x - 4)
Step 4) Simplify by combining "like-terms":
(x)(x) = x
1.
2
(x)(-4) = -4x
2.
(2)(x) = 2x
3.
(2)(-4) = -8
4.
x + 2x - 4x - 8
2
The pink terms can be combined
x - 2x - 8
2
Our expanded trinomial matches the one we were given to start, so is correct.
(x + 2)(x - 4)
Example 2) Factor this trinomial:
6x - 21x + 9
2
(x + a)(x + b)
_
_
Step 1) Take the GCF of 3 out first
6x - 21x + 9
2
3
3(2x - 7x + 3)
2
Step 2) Use the cross multiplication method to find a and b values that add up to make -7 and produce +3:
2x
a
1x
b
2x
-1
1x
-3
(-1)*(-3) = +3
1x(-1) = -1x
(-1x) + (-6x) = -7x
2x(-3) = -6x
Combine like terms and we get -7x (thats what we were looking for)
(-1) times (-3) produce +3
(thats what we were looking for)
Step 3) So, -1 and -3 are our a and b values. Now write them in the correct form:
(2x - 1)(x - 3)
Factoring Polynomials
Factoring Special Trinomials
Multiplying Polynomials
Multiplying Special Polynomials
Difference of Squares
INEQUALITIES
Solving Inequalities
Level 1
Level 3
Level 2
Level 4
SYSTEMS OF EQUATIONS
Solving by Substitution
Example 1) Solve this linear system
2x - 4y = 7
4x + y = 5
Step 1) Solve equation for y
4x + y = 5
-4x
-4x
Step 2) Substitute y = 5 - 4x in equation .
y = 5 - 4x
2x - 4y = 7
2x - 4(5 - 4x) = 7
Simplify, then solve for x
2x - 20 + 16x = 7
18x = 27
x = 1.5
Step 3) Substitute x = 1.5 in equation .
4x + y = 5
4(1.5)+ y = 5
6 + y = 5
y = -1
So x = 1.5
y = -1
Solving by Elimination
Example 1) Solve this linear system
3x - 4y = 7
5x - 6y = 8
Step 1) Look for terms with equal coefficients; since there are none, we can multiply each equation to make one.
Consider the x-terms. (3x) & (5x)
Their LCM is 15.
To make these terms equal, multiply equation by 5, and multiply equation by 3.
5 x equation : 5(3x - 4y = 7) 15x - 20y = 35
3 x equation : 3(5x - 6y = 8) 15x - 18y = 24
Step 2) Subtract equation from equation to eliminate x, and solve for y.
15x - 20y = 35
- (15x - 18y = 24)
- 20y - (-18y) = 35 - 24
- 2y = 11
y = - 5.5
y = - 5.5
Step 3) Substitute y = - 5.5 in equation and solve for x.
3x - 4y = 7
3x - 4(- 5.5) = 7
3x + 22 = 7
3x = - 15
x = - 5
x = - 5