The Language of Mathematics
Variable: A letter that represents an unknown number (usually x, y, or n)
Constant: A number that doesn't change (like 5, -3, or π)
Expression: A combination of variables, numbers, and operations (like 3x + 5)
Coefficient: The number multiplied by a variable (in 3x, the coefficient is 3)
Term: A single number, variable, or number times a variable (3x and 5 are terms)
Yahuah created a universe of order and pattern. Algebra helps us describe these patterns using symbols!
| Words | Expression |
|---|---|
| A number plus 7 | x + 7 |
| 5 times a number | 5n |
| A number divided by 3 | n ÷ 3 or n/3 |
| 8 less than a number | x - 8 |
| Twice a number plus 4 | 2x + 4 |
To evaluate an expression, substitute a number for the variable.
Answer: 17
The Tabernacle had 48 boards total. Let b represent the number of boards on one side. Write an expression for: "If three sides have the same number of boards and one side has 6 boards, write an expression for the total."
Expression:
If each of the three equal sides has 14 boards, check if this equals 48:
Parentheses → Exponents → Multiplication/Division (left to right) → Addition/Subtraction (left to right)
"Please Excuse My Dear Aunt Sally" or think of it as Yahuah's orderly creation - everything in its proper place!
An equation is like a balance scale. Whatever you do to one side, you must do to the other side to keep it balanced!
To solve: Use inverse (opposite) operations to isolate the variable.
| Operation | Inverse |
|---|---|
| Addition (+) | Subtraction (-) |
| Subtraction (-) | Addition (+) |
| Multiplication (×) | Division (÷) |
| Division (÷) | Multiplication (×) |
Step 1: Undo addition or subtraction (move constants away from the variable)
Step 2: Undo multiplication or division (isolate the variable)
Think: Reverse PEMDAS - undo operations in reverse order!
The Israelites were commanded to give a tithe (1/10) plus 5 shekels to the priests. If the total given was 17 shekels, how much was the original amount before tithing?
Let x = original amount. Write and solve the equation:
Equation: = 17
Solution: x = shekels
| Symbol | Meaning | Example |
|---|---|---|
| < | Less than | x < 5 (x is less than 5) |
| > | Greater than | x > 3 (x is greater than 3) |
| ≤ | Less than or equal to | x ≤ 7 (x is 7 or less) |
| ≥ | Greater than or equal to | x ≥ 2 (x is 2 or more) |
When you multiply or divide by a negative number, you must flip the inequality sign!
Example: -2x > 6 → x < -3 (sign flipped!)
Solution: All numbers less than 3
Solution: All numbers -3 or less
The coordinate plane has two axes:
Each point is written as an ordered pair (x, y)
The origin is at (0, 0)
To graph y = 2x + 1:
| x | y = 2x + 1 | Point |
|---|---|---|
| -1 | 2(-1) + 1 = -1 | (-1, -1) |
| 0 | 2(0) + 1 = 1 | (0, 1) |
| 1 | 2(1) + 1 = 3 | (1, 3) |
| 2 | 2(2) + 1 = 5 | (2, 5) |
Plot these points and connect with a straight line!
For y = x + 3:
| x | y |
|---|---|
| -2 | |
| 0 | |
| 2 |
For y = 2x - 1:
| x | y |
|---|---|
| -1 | |
| 0 | |
| 1 | |
| 2 |
slope = m = (y₂ - y₁) / (x₂ - x₁) = rise / run
y = mx + b
| Type | Appearance | Value |
|---|---|---|
| Positive | Line goes up (left to right) | m > 0 |
| Negative | Line goes down (left to right) | m < 0 |
| Zero | Horizontal line | m = 0 |
| Undefined | Vertical line | Cannot divide by 0 |
Slope = 2 (rises 2 for every 1 to the right)
A system of equations is two or more equations with the same variables. The solution is the point where the lines intersect!
y = 2x + 1
y = x + 4
Solution: (3, 7)
x^a × x^b = x^(a+b) | x^a ÷ x^b = x^(a-b) | (x^a)^b = x^(ab)
x^0 = 1 | x^(-n) = 1/x^n
1) n + 6 or x + 6 2) 4n - 2 or 4x - 2 3) 17 4) 21 5) 15
Word Problem: 3b + 6; 3(14) + 6 = 42 + 6 = 48 ✓
1) 17 2) 20 3) 18 4) 8 5) 12 6) 5
1) x = 6 2) n = 19 3) x = 9 4) y = 32 5) x = 37 6) n = 9
1) x = 5 2) n = 9 3) x = 20 4) y = 6 5) n = 24
Word Problem: (x/10) + 5 = 17; x = 120 shekels
1) x = 4 2) n = 5 3) x = 4 4) y = 11 5) x = 4
1) x < 7 2) n ≥ 5 3) x < -5 4) y ≤ 3
y = x + 3: (−2, 1), (0, 3), (2, 5)
y = 2x − 1: (−1, −3), (0, −1), (1, 1), (2, 3)
1) m = 3 2) m = -1 3) m = 3 4) b = 5
1) (3, 5) 2) (4, 9)
1) x⁷ 2) y⁴ 3) x⁶ 4) 5x + 3 5) 2x² - 4x