Linear Equations, Slope, and Arithmetic Sequences: College Algebra Study Notes

Equations & Inequalities

Graphing Linear Equations

Linear equations can be graphed by identifying their slope and intercepts. The general form of a linear equation is Ax + By = C or y = mx + b, where m is the slope and b is the y-intercept.

• Standard Form: $Ax + By = C$

• Slope-Intercept Form: $y = mx + b$

• Vertical Lines: $x = a$ (undefined slope)

• Horizontal Lines: $y = b$ (zero slope)

• To graph:

  1. Solve for y to get slope-intercept form if needed.

  2. Plot the y-intercept.

  3. Use the slope to find another point.

  4. Draw the line through the points.

• Example: Graph $6x + 5y = -20$

  • Solve for y: $5y = -6x - 20 \ y = -\frac{6}{5}x - 4$

  • Slope: $-\frac{6}{5}$, y-intercept: $-4$

Finding x- and y-intercepts

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 x-intercept: Set $y = 0$ and solve for $x$.

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

• Example: $8x + 6y = 12$

  • x-intercept: $8x = 12 \ x = 1.5$

  • y-intercept: $6y = 12 \ y = 2$

Graphs of Equations

Identifying Solutions to Linear Equations

A point $(x, y)$ is a solution to a linear equation if substituting $x$ and $y$ into the equation makes it true.

• Example: Is $(0, 3)$ a solution to $-5x - 2y = 6$?

  • Substitute: $-5(0) - 2(3) = -6$

  • Result: $-6 = 6$ (False), so not a solution.

Functions

Rate of Change and Slope

The slope of a line measures its steepness and is calculated as the ratio of the change in y to the change in x between two points.

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

• Rate of Change: In applications, slope represents the rate at which one variable changes with respect to another.

• Example: Mary stuffs 32 envelopes in 4 minutes and 72 envelopes in 9 minutes.

  • Rate: $\frac{72 - 32}{9 - 4} = \frac{40}{5} = 8$ envelopes per minute

Types of Slope

• Positive Slope: Line rises left to right.

• Negative Slope: Line falls left to right.

• Zero Slope: Horizontal line.

• Undefined Slope: Vertical line.

Systems of Equations & Matrices

Writing Equations of Lines

Lines can be written in point-slope form or slope-intercept form using a point and the slope.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• Example: Through $(3,2)$ and $(8,-5)$

  • Slope: $m = \frac{-5 - 2}{8 - 3} = \frac{-7}{5}$

  • Point-slope: $y - 2 = -\frac{7}{5}(x - 3)$

  • Slope-intercept: $y = -\frac{7}{5}x + \frac{21}{5} + 2$

Parallel and Perpendicular Lines

• Parallel lines: Same slope, different y-intercepts.

• Perpendicular lines: Slopes are negative reciprocals.

• Example: Line parallel to $x = 9$ through $(-2,4)$ is $x = -2$ (vertical line).

Sequences, Series, & Induction

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers with a constant difference between consecutive terms.

• General Term: $a_n = a_1 + d(n-1)$

• Common Difference: $d = a_{n+1} - a_n$

• Example: Sequence: 6, -1, -8, -15...

  • Common difference: $-1 - 6 = -7$

  • Explicit formula: $a_n = 6 + (-7)(n-1)$

• Finding a Specific Term:

  • Given $a_1 = 137$, $a_{40} = 389$, find $d$:

    • $d = \frac{389 - 137}{40 - 1} = \frac{252}{39} \approx 6.46$

  • Find $a_{138}$: $a_{138} = 137 + 6.46 \times (138 - 1)$

Identifying Arithmetic Sequences

• Check if the difference between consecutive terms is constant.

• Example: 3, 8, 13, 18... (difference is 5, so arithmetic)

Additional Info

• Some problems involve interpreting tables and graphs to find slope and intercepts.

• Questions also cover determining if statements about lines are always, sometimes, or never true.

HTML Table: Arithmetic Sequence Example

Term (n)	Value (a_n)	Difference (d)
1	6	-7
2	-1	-7
3	-8	-7
4	-15	-7