Further Solving of Equations and Inequalities

1. Simultaneous Equations

Two linear equations with two unknowns (usually $x$ and $y$) that must be solved simultaneously.

Methods of Solution

  • Elimination: Multiply equations so that coefficients of one variable are the same (or opposites), then add or subtract the equations to eliminate that variable.

    Example: Solve $2x + 3y = 13$ and $x - 2y = -4$

    1. Multiply the second equation by 2 to match $x$ coefficients $\rightarrow 2x - 4y = -8$
    2. Subtract the new equation from the first $\rightarrow (2x + 3y) - (2x - 4y) = 13 - (-8)$
    3. Simplify and solve for $y \rightarrow 7y = 21 \rightarrow y = 3$
    4. Substitute $y = 3$ back into $x - 2y = -4 \rightarrow x - 2(3) = -4 \rightarrow x = 2$
    5. Result: $(2, 3)$

  • Substitution: Rearrange one equation to express one variable in terms of the other (e.g., $x = \dots$), then substitute this expression into the second equation.

    Example: Solve $y = 2x + 1$ and $3x + 2y = 16$
    Steps:

    1. Substitute the expression for $y$ into the second equation $\rightarrow 3x + 2(2x + 1) = 16$
    2. Expand the brackets $\rightarrow 3x + 4x + 2 = 16$
    3. Simplify and solve for $x \rightarrow 7x = 14 \rightarrow x = 2$
    4. Substitute $x = 2$ back into $y = 2x + 1 \rightarrow y = 2(2) + 1 \rightarrow y = 5$
    5. Result: $(2, 5)$

2. Linear Inequalities

Inequalities describe a range of values rather than a single value.

Solving Inequalities

Solve as you would a linear equation, with one critical rule: When multiplying or dividing by a negative number, reverse the inequality sign.

  • Example: $-2x < 6 \implies x > -3$

Representation

  • Number Line:
    • Open circle $\circ$ for $<$ or $>$.
    • Closed circle $\bullet$ for $\le$ or $\ge$.
  • Regions in a Plane:
    • Solid line for $\le$ or $\ge$.
    • Dashed line for $<$ or $>$.
    • Shade the region that satisfies the inequality.

Inequality Region Plot

3. Quadratic Methods

Solving equations of the form $ax^2 + bx + c = 0$.

Completing the Square

Transform $x^2 + bx + c$ into the form $(x + p)^2 + q$.

  1. Take half of $b$ for the bracket: $(x + \frac{b}{2})^2$.
  2. Subtract $(\frac{b}{2})^2$ and add $c$: $(x + \frac{b}{2})^2 - (\frac{b}{2})^2 + c$.

The Quadratic Formula

Used when factorisation is difficult or impossible. $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ The term $b^2 - 4ac$ is the discriminant:

  • $> 0$: Two distinct real roots.
  • $= 0$: One repeated real root.
  • $< 0$: No real roots.

4. Quadratic Factorisation

Factorising $ax^2 + bx + c$ where $a \neq 1$.

Method: Split the Middle Term

  1. Find two numbers that multiply to $ac$ and add to $b$.
  2. Rewrite $bx$ using these two numbers.
  3. Factorise by grouping.

Example: $2x^2 + 7x + 3$

  • $ac = 6, b = 7 \rightarrow$ Numbers are $6$ and $1$.
  • $2x^2 + 6x + x + 3$
  • $2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)$

5. Algebraic Functions

Notation and Application

  • Function Notation: $f(x)$ denotes the output of function $f$ for input $x$.
  • Domain: The set of all possible input values ($x$).
  • Range: The set of all possible output values ($f(x)$).
  • Inverse Function $f^{-1}(x)$: The function that “reverses” the action of $f(x)$.
  • Composite Function $gf(x)$: Apply $f$ first, then apply $g$ to the result.

Function Mapping Diagram