Equations
Solving simple equations
Examples
- \(3x+10=x+4\)
- \(\frac{z}{z-5}+\frac{1}{3}=\frac{-5}{5-z}\)
- \(Y=C+I\)
- Fundamental rule for equation solving: Always exclude any value that makes an equation undefined.
- What about the examples above?
Invariance of equations
- An equation is invariant to the addiction (or subtraction) of the same number
- An equation is also invariant to the multiplication (division) by the same number \(n\neq 0\)
Three examples
- \(6p-\frac{1}{2}(2p-3)=3(1-p)-\frac{7}{6}(p+2)\)
- \(\frac{x+2}{x-2}-\frac{8}{x^2-2x}=\frac{2}{x}\)
- 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
- \(y=10x\)
- \(y=3x+4\)
- \(y=-\frac{8}{3}x-\frac{7}{2}\)
General form (simple linear relationship)
- \(y=ax+b\). Where \(a\) is the slope parameter and \(b\) is the intercept or slope parameter
- Draw the the simple linear relationship in a graph
Example: the basic macroeconomic model
- Output equation: \(Y=C+\bar{I}\)
- Consumption equation: \(C=a+bY\)
- What is the economic interpretation of these equations?
- What is the solution to the model? What is output as a function of investment?
A numerical example
- \(Y=C+100\) and \(C=500+.8Y\)
- Structural (separate equations) and reduced-form (one equation) of the model
Example: Money demand
- \(M=\alpha Y+\beta(r-\lambda)^{-\delta}\)
- Solve the equation for \(r\)
- USA data for the period 1929-1952: \(\alpha=.14\), \(\beta=76.03\), \(\lambda=2\), \(\delta=.84\)
Quadratic equations
The general form
- \(ax^2+bx+c=0\), where \((a\neq 0)\)
- Solving the general form for \(x\)
- What if \(b=0\)?
- \(x^2+c=0\). Solving for \(x\) yields: \(ax^2+c=0\Longleftrightarrow x=\pm \sqrt{-\frac{c}{a}}\). What is a requirement for this equation in the realm of the real numbers? \(\frac{c}{a}\leq 0\).
Examples
- \(5x^2-8x=0\)
- \(x^2-4=0\)
- \(x^2+3=0\)
Harder cases: completing the square
- All three parameters, \(a, b, \& c\), are different from zero
- Take the following example: \(12x^2-16x-3=0\)? How would you proceed to solve it?
- Consider the following equation: \(x^2+8x-9=0\). One way of solving this is by completing the square.
- \(x^2+8x=9\)
- \(x^2+8x+16=16+9 \)
- \(x^2+8x+16=25\)
- \((x+4)^2=25\)
- \((x+4)^2-5^2=0\)
- \((x+4+5)(x+4-5)=0\)
- \((x+9)(x-1)=0\)
- We thus get the solutions: \(x=-9\) and \(x=1\)
The general case
- \(ax^2+bx+c=0\)
- \(a(x^2+\frac{b}{a}x+\frac{c}{a})=0\)
- Given that, by assumption, \(a\neq 0\), \(x^2+\frac{b}{a}x=-\frac{c}{a}\)
- We now complete the square: \(x^2+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2\)
- \((x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\)
- \(x+\frac{b}{2a}=\frac{\pm\sqrt{b^2-4ac}}{2a}\)
- \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
- The equation isn 7. is referred to as the quadratic formula
Numerical examples
- \(2x^2-2x-40=0\)
- \(\frac{1}{3}x^2+\frac{2}{3}x-\frac{14}{3}=0\)
- \(-2x^2+40x-600=0\)
- Should you be a "quadratic formula fanatic"?
Linear equations in two unknowns
Two examples
- \(2x+3y=18\)
- \(3x-4y=-7\)
- We usually solve this for \(y\)
Nonlinear equations
Numerical examples
- \(x^3\cdot \sqrt{x+2}=0\)
- \(x(y+3)(z^2+1)\sqrt{w-3}=0\)
- \(x^2-3x^3=0\)
- \(\frac{1-K^2}{\sqrt{1+K^2}}=0\)
- \(\frac{45+6r-3r^2}{(r^4+2)^{3/2}}=0\)