📐 The Language of Algebra
🔍 Connecting to observations: Remember the balance scale, age puzzle, shopping bill, number patterns, and garden area. Each demonstrates algebraic concepts in action!
1. Variables and Constants
Variable: A symbol (usually a letter) that represents an unknown number.
Constant: A fixed value that doesn't change.
x, y, a, b, c = variables
2, 5, -3, ½, π = constants
📌 Connecting to Observation 1 (Balance Scale):
We let a = weight of one apple (variable). The constant 100g was fixed. The equation 3a = 2a + 100 uses both variables and constants.
💡 Tip: Think of variables as empty boxes waiting to be filled with numbers!
2. Expressions vs Equations
| Expression |
Equation |
| 2x + 3 |
2x + 3 = 7 |
| No equals sign |
Has equals sign (=) |
| Can be simplified |
Can be solved |
| Example: 3a + 2b |
Example: 3a + 2b = 11 |
📌 From Observation 3 (Shopping):
Expressions: 3p + 2n (cost of 3 pens and 2 notebooks)
Equation: 3p + 2n = 11 (total cost equals $11)
3. Solving Linear Equations
Golden Rule: Whatever you do to one side, do to the other!
Steps to solve:
1. Simplify both sides
2. Add/subtract terms to get variable alone
3. Multiply/divide to solve
4. Check your answer
📌 Example: 3x + 5 = 20
- Subtract 5 from both sides: 3x = 15
- Divide both sides by 3: x = 5
- Check: 3(5) + 5 = 15 + 5 = 20 ✓
📌 From Observation 1 (Balance):
3a = 2a + 100
Subtract 2a: a = 100
⚠️ Common Mistake: Forgetting to perform the same operation on BOTH sides of the equation!
4. Systems of Equations
When we have two or more equations with the same variables, we solve them together.
Two methods:
1. Substitution
2. Elimination
📌 From Observation 3 (Shopping):
3p + 2n = 11 ...(1)
1p + 3n = 13 ...(2)
Method 1 - Substitution:
From (2): p = 13 - 3n
Substitute into (1): 3(13 - 3n) + 2n = 11
39 - 9n + 2n = 11
39 - 7n = 11
-7n = -28
n = 4, then p = 13 - 12 = 1
Method 2 - Elimination:
Multiply (2) by 3: 3p + 9n = 39
Subtract (1): (3p + 9n) - (3p + 2n) = 39 - 11
7n = 28, n = 4, then p = 1
💡 Tip: Choose the method that looks easier for your problem!
5. Quadratic Equations
ax² + bx + c = 0
where a ≠ 0
Quadratic Formula:
x = [-b ± √(b² - 4ac)] / 2a
📌 From Observation 5 (Garden):
w(w + 5) = 84
w² + 5w - 84 = 0
Here a = 1, b = 5, c = -84
Using quadratic formula:
w = [-5 ± √(25 - 4×1×-84)] / 2
w = [-5 ± √(25 + 336)] / 2
w = [-5 ± √361] / 2
w = [-5 ± 19] / 2
w = 7 or w = -12
Since width can't be negative, w = 7m
🧠 The Discriminant (b² - 4ac):
• If > 0: Two real solutions
• If = 0: One real solution
• If < 0: No real solutions
6. Inequalities
<, >, ≤, ≥
Rules:
• Adding/subtracting same number preserves inequality
• Multiplying/dividing by positive number preserves inequality
• Multiplying/dividing by negative number REVERSES inequality
📌 Example: -2x < 6
Divide both sides by -2 (remember to flip sign!):
x > -3
⚠️ Critical Rule: When multiplying or dividing by a negative number, FLIP the inequality sign!
7. Polynomials
| Type |
Form |
Example |
| Monomial |
axⁿ |
3x² |
| Binomial |
axⁿ + bxᵐ |
2x + 5 |
| Trinomial |
axⁿ + bxᵐ + cxᵏ |
x² + 5x + 6 |
Operations:
• Adding: Combine like terms
• Multiplying: Distribute
• Factoring: Reverse of multiplication
📌 Factoring Example:
x² + 5x + 6 = (x + 2)(x + 3)
Check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓
📊 Quick Reference
| Concept |
Rule/Formula |
Example |
| Commutative |
a + b = b + a |
3 + 5 = 5 + 3 |
| Associative |
(a + b) + c = a + (b + c) |
(2 + 3) + 4 = 2 + (3 + 4) |
| Distributive |
a(b + c) = ab + ac |
3(x + 2) = 3x + 6 |
| Zero Product |
If ab = 0, then a = 0 or b = 0 |
(x - 2)(x + 3) = 0 → x = 2 or x = -3 |
| Exponent Rules |
xᵃ · xᵇ = xᵃ⁺ᵇ |
x² · x³ = x⁵ |
⚠️ Common Mistakes to Avoid
Mistake 1: Forgetting to distribute negative signs
✗ 3 - (x + 2) = 3 - x + 2
✓ 3 - (x + 2) = 3 - x - 2 = 1 - x
Mistake 2: Incorrectly canceling terms
✗ (x + 2)/2 = x (dividing only x, not the +2)
✓ (x + 2)/2 = x/2 + 1
Mistake 3: Forgetting to flip inequality sign when multiplying by negative
✗ -2x < 6 → x < -3
✓ -2x < 6 → x > -3
Mistake 4: Losing solutions when taking square roots
✗ x² = 9 → x = 3
✓ x² = 9 → x = ±3