Algebra Review

Algebra Review

Key Concepts: Fractions (addition, subtraction, multiplication, division), like terms, exponents (multiplication, division, power of a power), distribution, FOIL method, solving linear equations (one-step, multi-step, with variables on both sides, with parentheses, with fractions, with decimals), cross multiplication.

Fractions

• Adding/Subtracting Fractions: Requires a common denominator. To achieve this, multiply each fraction by a form of 1 (e.g., 5/5, 4/4) using the denominator of the other fraction.
  • Example: 3/4 + 2/5 = (3/4 * 5/5) + (2/5 * 4/4) = 15/20 + 8/20 = 23/20
• Multiplying Fractions: Multiply straight across the numerators and denominators. Simplify the resulting fraction if possible by factoring and canceling common factors.
  • Example: 7/5 * 4/3 = 28/15
  • Simplification Example: 3/5 * 6/4 = 18/20 = (9 * 2) / (10 * 2) = 9/10
• Dividing Fractions: Use the "keep, change, flip" method. Keep the first fraction, change division to multiplication, and flip (reciprocal) the second fraction. Then, multiply as usual. Simplify before multiplying if possible.
  • Example: 36/52 ÷ 27/65 = 36/52 * 65/27
  • Simplification: (9 * 4)/(13 * 4) * (13 * 5)/(9 * 3) = (<s>9</s> * <s>4</s>)/(<s>13</s> * <s>4</s>) * (<s>13</s> * 5)/(<s>9</s> * 3) = 5/3

Combining Like Terms

• Like Terms: Terms with the same variable raised to the same power. Only like terms can be added or subtracted.
  • Example: 5x + 3x² - 7x + 4x³ + 8x² = (5x - 7x) + (3x² + 8x²) + 4x³ = -2x + 11x² + 4x³
• Multiplying Unlike Terms: Unlike terms can be multiplied. Multiply the coefficients and add the exponents of the variables.
  • Example: 5x * 3x² = 15x³

Exponents

• Multiplying Variables with Exponents: Add the exponents.
  • Example: x⁴ * x⁷ = x^(4+7) = x¹¹
• Dividing Variables with Exponents: Subtract the exponents (top exponent minus bottom exponent).
  • Example: x⁷ / x⁴ = x^(7-4) = x³
• Power of a Power: Multiply the exponents.
  • Example: (x³)^4 = x^(3*4) = x¹²
• Negative Exponents: To make a negative exponent positive, move the variable to the denominator (or vice versa).
  • Example: x^(-3) = 1/x³
• Zero Exponent: Any non-zero number raised to the power of zero is equal to 1.
  • Example: (3xy²)^0 = 1

Distribution and FOIL

• Distribution: Multiply a single term (monomial) by each term inside parentheses.
  • Example: 3x * (5x - 4) = 15x² - 12x
• FOIL (First, Outer, Inner, Last): A method for multiplying two binomials.
  • Example: (2x + 3) * (3x - 5) = (2x * 3x) + (2x * -5) + (3 * 3x) + (3 * -5) = 6x² - 10x + 9x - 15 = 6x² - x - 15
• Binomial Squared: Remember that (a - b)² = (a - b)(a - b), and then use FOIL.
  • Example: (3x - 4)² = (3x - 4)(3x - 4) = 9x² - 12x - 12x + 16 = 9x² - 24x + 16
• Multiplying a Binomial by a Trinomial: Multiply each term in the binomial by each term in the trinomial, then combine like terms. Lining up like terms vertically can help with organization.

Solving Basic Equations

• Goal: Isolate the variable on one side of the equation.
• One-Step Equations: Use the opposite operation to isolate the variable.
  • Addition: x + 8 = 15 => x = 15 - 8 => x = 7
  • Subtraction: x - 4 = 12 => x = 12 + 4 => x = 16
  • Multiplication: 3x = 15 => x = 15 / 3 => x = 5
  • Division: x / 6 = 4 => x = 4 * 6 => x = 24
• Fractional Coefficient: Multiply by the reciprocal of the fraction.
  • Example: (2/3)x = 8 => x = 8 * (3/2) => x = 24/2 => x = 12
• Multi-Step Equations: Use the order of operations in reverse to isolate the variable.
  • Example: 3x + 5 = 11 => 3x = 11 - 5 => 3x = 6 => x = 6 / 3 => x = 2
• Variables on Both Sides: Move all variable terms to one side and all constant terms to the other side.
  • Example: 4x + 3 = 6x - 15 => 3 + 15 = 6x - 4x => 18 = 2x => x = 18 / 2 => x = 9
• Equations with Parentheses: Distribute first, then solve as usual.
  • Example: 3(2x - 4) = 5(3x + 2) - 3 => 6x - 12 = 15x + 10 - 3 => 6x - 12 = 15x + 7 => -12 - 7 = 15x - 6x => -19 = 9x => x = -19/9
• Equations with Fractions: Multiply both sides by the least common multiple of the denominators to eliminate fractions.
  • Example: (2/3)x + 5 = 8 => 3 * [(2/3)x + 5] = 3 * 8 => 2x + 15 = 24 => 2x = 9 => x = 9/2 = 4 1/2 = 4.5
• Equations with Decimals: Multiply both sides by a power of 10 (10, 100, 1000, etc.) to eliminate decimals.
  • Example: 0.8x + 0.3 = 0.5x + 1.4 => 10 * [0.8x + 0.3] = 10 * [0.5x + 1.4] => 8x + 3 = 5x + 14 => 3x = 11 => x = 11/3 = 3 2/3 ≈ 3.67
• Cross Multiplication: If two fractions are equal, cross multiply.
  • Example: x/5 = 8/9 => 9x = 40 => x = 40/9

Conclusion

This algebra review covers fundamental concepts necessary for success in algebra and higher-level math courses. Mastering these skills through practice and consistent effort is crucial for building a strong mathematical foundation.