Solving Linear Equations and Inequalities to Make Predictions

Solving Formulas by Substitution

• Formula Application: To find a specific variable value, substitute known data into a linear equation.

• Example (Apple Workforce Statistics):

  • Event $A$: African American ($7\%$ or $0.07$).

  • Event $H$: Hispanic.

  • Event $A \text{ OR } H$: African American or Hispanic ($18\%$ or $0.18$).

  • Since the events are disjoint, we use $P(A \text{ OR } H) = P(A) + P(H)$.

  • Substitution: $0.18 = 0.07 + P(H)$.

  • Result: $P(H) = 0.11$.

Evaluating Summation Notation

• Expected Value Calculation: Summing the products of values and their respective probabilities.

• Formula: $\mu = \sum x_i P(x_i)$.

• Data Set:

  • $x_1 = 0, x_2 = 1, x_3 = 2, x_4 = 3, x_5 = 4$.

  • $P(x_1) = 0.0625, P(x_2) = 0.25, P(x_3) = 0.375, P(x_4) = 0.25, P(x_5) = 0.0625$.

• Calculation: $0(0.0625) + 1(0.25) + 2(0.375) + 3(0.25) + 4(0.0625) = 2.0$.

Solving for Fahrenheit and Celsius

• Variable Manipulation: Solving the formula $C = \frac{5}{9}(F - 32)$ for $F$.

• Resulting Formula: $F = \frac{9}{5}C + 32$.

• Equivalencies:

  • $10^\circ\text{C} = 50^\circ\text{F}$

  • $15^\circ\text{C} = 59^\circ\text{F}$

  • $20^\circ\text{C} = 68^\circ\text{F}$

  • $25^\circ\text{C} = 77^\circ\text{F}$

  • $30^\circ\text{C} = 86^\circ\text{F}$

Squaring Principal Square Roots

• Definition: For any nonnegative $x$, $(\sqrt{x})^2 = x$.

• Application (Margin of Error): To solve for the sample size $n$ in the formula $E = \frac{z \cdot \sigma}{\sqrt{n}}$:

  1. Isolate the radical: $\sqrt{n} = \frac{z \cdot \sigma}{E}$.

  2. Square both sides: $n = (\frac{z \cdot \sigma}{E})^2$.

Graphing by Solving for y

• Slope-Intercept Form: Convert linear equations into $y = a + bx$ to identify the slope and intercept.

• Example: $2x + 3y = 9$.

  • Isolating $y$: $3y = -2x + 9 \rightarrow y = -\frac{2}{3}x + 3$.

  • Graphing Attributes:

    • The $y$-intercept is $(0, 3)$.

    • The slope is $-\frac{2}{3}$.