Elementary Algebra

 

1. Introduction to Algebra

  • Definition: Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. The symbols (often letters) represent numbers and quantities in formulas and equations.
  • Expressions vs. Equations:
    • Expression: A combination of numbers, variables, and operations (e.g., 3x+5).
    • Equation: A statement that two expressions are equal (e.g., 3x+5=11).

2. Basic Concepts

  • Variables: Symbols (typically letters) that represent unknown values. Common examples include xx, yy, and zz.
  • Constants: Fixed values, like numbers (e.g., 2, -3, 4.5).
  • Coefficients: Numbers that multiply a variable (e.g., in
    3x
    , 3 is the coefficient).
  • Terms: Individual parts of an expression, separated by addition or subtraction (e.g., in 3x+53x + 5
    and 5 are terms).
  • Polynomials: Expressions that consist of multiple terms (e.g., 2x2+3x+4).

3. Operations on Algebraic Expressions

  • Addition/Subtraction: Combine like terms (e.g., 3x+2x=5x).
  • Multiplication: Use distributive property (e.g., 2(x+3)=2x+62(x + 3) = 2x + 6).
  • Division: Divide coefficients and subtract exponents for similar bases (e.g., 6x23x=2x\frac{6x^2}{3x} = 2x).

4. Solving Linear Equations

  • Linear Equation: An equation of the first degree (e.g., ax+b=0).
  • Solution Steps:
    1. Simplify both sides (if needed).
    2. Isolate the variable (e.g., move terms involving xx to one side and constants to the other).
    3. Solve for the variable (e.g., x=bax = \frac{-b}{a}).
  • Examples:
    • 2x+3=72x + 3 = 7 → 2x=42x = 4x=2x = 2
    • 3x4=2x+13x - 4 = 2x + 1 → x=5x = 5

5. Inequalities

  • Definition: Similar to equations but involve inequalities (<<, \leq, >>, \geq).
  • Solving Steps:
    1. Simplify both sides.
    2. Isolate the variable.
    3. Solve, remembering to reverse the inequality sign when multiplying or dividing by a negative number.
  • Graphical Representation: Inequalities can be represented on a number line.

6. Factoring

  • Definition: Breaking down a complex expression into a product of simpler expressions.
  • Common Methods:
    • Common Factor: Factor out the greatest common factor (GCF) (e.g., 6x+9=3(2x+3)6x + 9 = 3(2x + 3)
    • Difference of Squares: a2b2=(a+b)(ab)a^2 - b^2 = (a + b)(a - b)
    • Quadratic Trinomials: ax2+bx+c can sometimes be factored into (mx+n)(px+q)(mx + n)(px + q)
  • Examples:
    • x29=(x+3)(x3)x^2 - 9 = (x + 3)(x - 3)
    • x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)

7. Quadratic Equations

  • Definition: Equations of the form ax2+bx+c=0ax^2 + bx + c = 0.
  • Solution Methods:
    • Factoring: (if factorable) x2+5x+6=(x+2)(x+3)=0x^2 + 5x + 6 = (x + 2)(x + 3) = 0
    • or x = −3
    • Quadratic Formula: x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
    • Completing the Square: Transform the equation into a perfect square trinomial and solve.

8. Exponents and Powers

  • Rules:
    • am×an=am+na^m \times a^n = a^{m+n}
    • aman=amn
    • (am)n=amn
    • an=1ana^{-n} = \frac{1}{a^n}
    • a0=1a^0 = 1 (where a0).
  • Examples:
    • 23×22=25=.
    • x5x2=x3\frac{x^5}{x^2} = x^{3}

9. Radicals and Rational Exponents

  • Radicals: Expressions involving roots, such as x\sqrt{x} (square root).
  • Properties:
    • a×b=ab\sqrt{a} \times \sqrt{b} = \sqrt{ab}.
    • ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
  • Rational Exponents: amn=amna^{\frac{m}{n}} = \sqrt[n]{a^m}
    • Example: x12=xx^{\frac{1}{2}} = \sqrt{x}

10. Functions

  • Definition: A relationship between two sets where each input (x) has exactly one output (y).
  • Notation: f(x), where ff is the function, and xx is the input.
  • Linear Function: f(x)=mx+bf(x) = mx + b (straight-line graph).
  • Quadratic Function: f(x)=ax2+bx+cf(x) = ax^2 + bx + c (parabola-shaped graph).

11. Systems of Equations

  • Definition: A set of two or more equations with the same variables.
  • Solution Methods:
    • Substitution: Solve one equation for one variable and substitute into the other.
    • Elimination: Add or subtract equations to eliminate one variable.
    • Graphical Method: Graph both equations and find the point(s) of the intersection.
  • Example:
    • x+y=3 and 2xy=12x - y = 1
  • Steps to Solve:
    1. Define the variables.
    2. Write an equation based on the problem.
    3. Solve the equation.
    4. Interpret the solution in the context of the problem.
  • Examples:
    • If a number is doubled and then increased by 5, the result is 15. What is the number? (Equation: 2x+5=152x + 5 = 15).

13. Rational Expressions and Equations

  • Rational Expressions: Fractions where the numerator and/or denominator is a polynomial (e.g., 2x+3x1\frac{2x + 3}{x - 1}).
  • Operations:
    • Addition/Subtraction: Find a common denominator.
    • Multiplication: Multiply numerators together and denominators together.
    • Division: Multiply by the reciprocal of the divisor.
  • Solving Rational Equations: Clear fractions by multiplying by the least common denominator (LCD), then solve the resulting equation.

14. Absolute Value

  • Definition: The distance of a number from zero on a number line, is always non-negative.
    • x=x|x| = x if x0x \geq 0.
    • x=x|x| = -x if x<0x < 0.
  • Absolute Value Equations: Solve by considering both the positive and negative cases.
    • Example: x3=5|x - 3| = 5 leads to
    •   x3=5x - 3 = 5 or x3=5x - 3 = -5
    • so x=8x = 8 or x=2x = -2

15. Conclusion

Elementary Algebra forms the foundation for more advanced mathematical concepts. These basic principles are crucial for success in higher-level mathematics, including geometry, calculus, and beyond.



Higher Maths

I have done my Masters in Mathematics, all my Subjects had vast calculations, theorem and formulas of Maths, so I have good knowledge of the subject. I have been tutoring students for more than 6 years now, and have good experience in solving the doubts and queries of students clearly with proper explanations. I have worked with many online tutoring sites in the past, where I received overwhelming responses from students for my teaching. They loved my tutoring style and the way I cleared all their doubts and queries step by step. So I have good experience with online tutoring and I will work wholeheartedly to give my best to students.

Post a Comment

Previous Post Next Post