Linear Equations and Slopes Practice

Vocabulary and key practice problems covering the calculation of slope, slope-intercept form, point-slope form, and parallel and perpendicular line relationships.

Slope Formula

The mathematical formula used to find the steepness of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$, defined as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Slope-Intercept Form

The standard linear form represented by $y = mx + b$, where $m$ denotes the slope and $b$ represents the $y$-intercept.

Point-Slope Form

The equation of a straight line written in the form $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a specific coordinate point on the line.

Parallel Lines (Slopes)

Lines in the same coordinate plane that never intersect and possess equal slopes ($m_1 = m_2$).

Perpendicular Lines (Slopes)

Lines that intersect at a right angle ($90^\circ$) with slopes that are negative reciprocals of each other, satisfying the relationship $m_1 \times m_2 = -1$.

Slope through $(-14, 11)$ and $(5, 20)$

$\frac{9}{19}$, found by calculating $m = \frac{20 - 11}{5 - (-14)} = \frac{9}{19}$.

Slope through $(-12, 17)$ and $(-2, -11)$

$-\frac{14}{5}$, found by calculating $m = \frac{-11 - 17}{-2 - (-12)} = \frac{-28}{10} = -\frac{14}{5}$.

Slope through $(10, -1)$ and $(14, 10)$

$\frac{11}{4}$, found by calculating $m = \frac{10 - (-1)}{14 - 10} = \frac{11}{4}$.

Slope through $(11, 14)$ and $(-12, 4)$

$\frac{10}{23}$, found by calculating $m = \frac{4 - 14}{-12 - 11} = \frac{-10}{-23} = \frac{10}{23}$.

Slope-Intercept Equation through $(0, -4)$ and $(-1, -5)$

$y = x - 4$, derived using slope $m = \frac{-5 - (-4)}{-1 - 0} = 1$ and $y$-intercept $b = -4$.

Slope-Intercept Equation through $(-1, 5)$ and $(1, 3)$

$y = -x + 4$, derived using slope $m = \frac{3 - 5}{1 - (-1)} = -1$ and $y$-intercept $b = 4$.

Slope-Intercept Equation through $(-4, 0)$ and $(0, 2)$

$y = \frac{1}{2}x + 2$, derived using slope $m = \frac{2 - 0}{0 - (-4)} = \frac{1}{2}$ and $y$-intercept $b = 2$.

Slope-Intercept Equation through $(1, 1)$ Parallel to $y = -2x + 1$

$y = -2x + 3$, using slope $m = -2$ and solving for the intercept $b = 3$.

Slope-Intercept Equation through $(-3, 0)$ Parallel to $y = -\frac{1}{3}x + 1$

$y = -\frac{1}{3}x - 1$, using slope $m = -\frac{1}{3}$ and solving for the intercept $b = -1$.

Slope-Intercept Equation through $(-2, 1)$ Parallel to $y = \frac{1}{2}x - 4$

$y = \frac{1}{2}x + 2$, using slope $m = \frac{1}{2}$ and solving for the intercept $b = 2$.

Slope-Intercept Equation through $(-4, -5)$ Perpendicular to $y = \frac{1}{5}x + 4$

$y = -5x - 25$, using the perpendicular slope $m = -5$ (negative reciprocal of $\frac{1}{5}$) and coordinate point $(-4, -5)$.

Point-Slope Equation through $(1, 2)$ and $(0, 5)$

$y - 2 = -3(x - 1)$, with slope $m = \frac{5 - 2}{0 - 1} = -3$.