Equations

Solving simple equations

Examples

  1. \(3x+10=x+4\)
  2. \(\frac{z}{z-5}+\frac{1}{3}=\frac{-5}{5-z}\)
  3. \(Y=C+I\)

Invariance of equations

  1. An equation is invariant to the addiction (or subtraction) of the same number
  2. An equation is also invariant to the multiplication (division) by the same number \(n\neq 0\)

Three examples

  1. \(6p-\frac{1}{2}(2p-3)=3(1-p)-\frac{7}{6}(p+2)\)
  2. \(\frac{x+2}{x-2}-\frac{8}{x^2-2x}=\frac{2}{x}\)
  3. A firm manufactures a commodity that costs $20 per unit to produce. In addition, the firm has fixed costs of $2000. Each unit is sold for $75. How many units myst be sold if the firm is to meet a profit target of $14,500?

Equations with parameters

Numerical examples

  1. \(y=10x\)
  2. \(y=3x+4\)
  3. \(y=-\frac{8}{3}x-\frac{7}{2}\)

General form (simple linear relationship)

Example: the basic macroeconomic model

A numerical example

Example: Money demand

Quadratic equations

The general form

Examples

  1. \(5x^2-8x=0\)
  2. \(x^2-4=0\)
  3. \(x^2+3=0\)

Harder cases: completing the square

  1. \(x^2+8x=9\)
  2. \(x^2+8x+16=16+9 \)
  3. \(x^2+8x+16=25\)
  4. \((x+4)^2=25\)
  5. \((x+4)^2-5^2=0\)
  6. \((x+4+5)(x+4-5)=0\)
  7. \((x+9)(x-1)=0\)
  8. We thus get the solutions: \(x=-9\) and \(x=1\)

The general case

  1. \(ax^2+bx+c=0\)
  2. \(a(x^2+\frac{b}{a}x+\frac{c}{a})=0\)
  3. Given that, by assumption, \(a\neq 0\)\(x^2+\frac{b}{a}x=-\frac{c}{a}\)
  4. We now complete the square: \(x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2\)
  5. \((x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\)
  6. \(x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}\)
  7. \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Numerical examples

Linear equations in two unknowns

Two examples

Nonlinear equations

Numerical examples

  1. \(x^3\cdot \sqrt{x+2}=0\)
  2. \(x(y+3)(z^2+1)\sqrt{w-3}=0\)
  3. \(x^2-3x^3=0\)
  4. \(\frac{1-K^2}{\sqrt{1+K^2}}=0\)
  5. \(\frac{45+6r-3r^2}{(r^4+2)^{3/2}}=0\)