College Algebra Study Guide: Linear Equations, Functions, and Systems

Linear Equations and Their Graphs

Finding Intercepts

Intercepts are points where a graph crosses the axes. The x-intercept is where the graph crosses the x-axis (y=0), and the y-intercept is where it crosses the y-axis (x=0).

• To find the x-intercept: Set y = 0 and solve for x.

• To find the y-intercept: Set x = 0 and solve for y.

• Example: For $3x + 5y = 30$:

  • x-intercept: $y = 0 \Rightarrow 3x = 30 \Rightarrow x = 10$

  • y-intercept: $x = 0 \Rightarrow 5y = 30 \Rightarrow y = 6$

Graphing Linear Equations

To graph a linear equation, plot the intercepts or use the slope-intercept form.

• Slope-intercept formkm: $y = mx + b$, where m is the slope and b is the y-intercept.

• Graphing x = k: This is a vertical line at x = k.

Finding Slope

The slope of a line measures its steepness and is calculated from two points $(x_1, y_1)$ and $(x_2, y_2)$:

• Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: For points (2,1) and (8,4): $m = \frac{4-1}{8-2} = \frac{3}{6} = \frac{1}{2}$

Equation of a Line Through Two Points

To find the equation of a line through two points, first calculate the slope, then use point-slope or slope-intercept form.

• Point-slope form: $y - y_1 = m(x - x_1)$

• Example: Through $(-3,7)$ and $(11,11)$:

  • Slope: $m = \frac{11-7}{11-(-3)} = \frac{4}{14} = \frac{2}{7}$

  • Equation: $y - 7 = \frac{2}{7}(x + 3)$

Horizontal and Vertical Lines

• Horizontal line: $y = k$ (slope = 0)

• Vertical line: $x = k$ (undefined slope)

• Example: The equation of a horizontal line through $(2,5)$ is $y = 5$.

Functions and Their Properties

Function Notation and Evaluation

A function assigns each input exactly one output. Function notation: $f(x)$.

• Example: $q(x) = \frac{2x-3}{x+1}$

• To find $q(4)$: $q(4) = \frac{2\times4-3}{4+1} = \frac{8-3}{5} = \frac{5}{5} = 1$

• To find $q(0)$: $q(0) = \frac{2\times0-3}{0+1} = \frac{-3}{1} = -3$

• To find $q(-1)$: $q(-1) = \frac{2\times(-1)-3}{-1+1} = \frac{-2-3}{0}$ (undefined)

Domain and Range

The domain of a function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• For a graph, the domain is the interval covered by the x-axis, and the range is the interval covered by the y-axis.

• Example: For a graph that starts at $x = -2$ and ends at $x = 4$, the domain is $[-2, 4]$.

Piecewise Defined Functions

A piecewise function is defined by different expressions over different intervals.

• Example: $g(x) = \begin{cases} x^2 & x \leq 2 \\ -2 & x > 2 \end{cases}$

• Graph each piece over its specified interval.

Solving Systems of Equations

Solving by Substitution or Elimination

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Substitution method: Solve one equation for one variable, substitute into the other.

• Elimination method: Add or subtract equations to eliminate a variable.

• Example: Solve $x + y = 8$ and $y = 5x$:

  • Substitute $y = 5x$ into $x + y = 8$:

  • $x + 5x = 8 \Rightarrow 6x = 8 \Rightarrow x = \frac{4}{3}$

  • $y = 5x = 5 \times \frac{4}{3} = \frac{20}{3}$

Word Problems: Mixture and Cost Problems

Mixture and cost problems require setting up equations based on given conditions.

• Mixture problem: Combine two brands with different concentrations to achieve a desired mixture.

• Example: Brand A (25% dried fruit), Brand B (10% dried fruit), total 20 lbs at 19% dried fruit.

• Let $x$ = lbs of Brand A, $y$ = lbs of Brand B.

• Equations:

  • $x + y = 20$

  • $0.25x + 0.10y = 0.19 \times 20 = 3.8$

• Cost problem: Find quantities sold given total sales and prices.

• Example: White sweatshirts ($10.95$), yellow ($13.50$), total sold = 36, total sales = $435.

• Let $x$ = white, $y$ = yellow.

• Equations:

  • $x + y = 36$

  • $10.95x + 13.50y = 435$

Tables

Function Table: Evaluation

The table below summarizes the evaluation of $q(x)$ for selected values:

x	q(x)
4	1
0	-3
-1	undefined

Additional info:

• Some context and explanations have been expanded for clarity and completeness.

• Graphing instructions and interval notation for domain/range are standard topics in College Algebra.