This page contains a comprehensive set of 20 practice questions on slope and equations of straight lines. Topics include the slope formula, identifying vertical and horizontal lines, converting between standard and slope-intercept forms, determining parallel and perpendicular lines, and finding intersections. Complete step-by-step solutions are provided beneath each question.
Find, if possible, the slope of the line through \((\tfrac{1}{2}, \tfrac{3}{4})\) and \((-\tfrac{1}{3}, \tfrac{5}{4})\).
Use \( m = \dfrac{y_2 - y_1}{x_2 - x_1} \):
\[ m = \frac{\tfrac{5}{4} - \tfrac{3}{4}}{-\tfrac{1}{3} - \tfrac{1}{2}} = \frac{\tfrac{1}{2}}{-\tfrac{5}{6}} = \frac{1}{2} \cdot \left(-\frac{6}{5}\right) = -\frac{3}{5} \]Answer: \(-\frac{3}{5}\)
Find, if possible, the slope of the line through \((2, 5)\) and \((-4, 5)\).
Same y-coordinate → horizontal line, slope = 0.
Find, if possible, the slope of the line through \((-9, 4)\) and \((-9, -2)\).
Division by zero → slope is undefined (vertical line).
Line \(L\) passes through \((4, -5)\) and \((3, 7)\). Find the slope of any line perpendicular to \(L\).
Find the slope of the line \(2x + 4y = 10\).
Slope: \(-\frac{1}{2}\)
Write the slope-intercept equation of the line through \((2, 3)\) and \((4, 6)\).
Write the slope-intercept equation of the line with x-intercept \((3, 0)\) and y-intercept \((0, -5)\).
Equation: \(y = \tfrac{5}{3}x - 5\)
Write the slope-intercept equation of the line with slope \(-2\) through \((-4, -5)\).
Write the equation of the vertical line through \((3, 0)\).
Equation: \(x = 3\)
Write the equation of the horizontal line through \((7, -5)\).
Equation: \(y = -5\)
Find the line through \((-3, 5)\) and \((9, 10)\) in standard form \(Ax + By = C\), \(A > 0\).
Multiply by \(-1\):
\[ 5x - 12y = -75 \]Find the line parallel to \(3x + 6y = 5\) through \((1, 3)\), in slope-intercept form.
Find the line perpendicular to \(3x + 6y = 5\) through \((1, 3)\), in standard form.
Given slope \(-\tfrac{1}{2}\), so perpendicular slope \(= 2\):
\[ y - 3 = 2(x - 1) \implies y = 2x + 1 \] \[ -2x + y = 1 \implies 2x - y = -1 \]Answer: \(2x - y = -1\) (older keys gave only the slope-intercept form \(y = 2x + 1\))
Write the equation of the line parallel to \(x = 5\) through \((3, -10)\).
Parallel to a vertical line → vertical line through \(x = 3\).
Answer: \(x = 3\)
Write the equation of the line perpendicular to \(x = 2\) through \((2, -8)\).
Perpendicular to a vertical line → horizontal line through \(y = -8\).
Answer: \(y = -8\)
Write the line through the midpoint of \((2, 8)\) and \((0, 4)\), perpendicular to \(-3x + 6y = 5\).
Perpendicular slope \(= -2\):
\[ y - 6 = -2(x - 1) \implies y = -2x + 8 \]Classify \(2x - 3y = 8\) and \(-x + 4y = 2\): parallel, perpendicular, or neither.
Line 1: \(y = \tfrac{2}{3}x - \tfrac{8}{3}\) (\(m_1 = \tfrac{2}{3}\))
Line 2: \(y = \tfrac{1}{4}x + \tfrac{1}{2}\) (\(m_2 = \tfrac{1}{4}\))
Slopes are not equal and \(m_1 m_2 \neq -1\).
Answer: Neither
Classify \(2x = 8\) and \(-3y = 15\).
\(x = 4\) (vertical) and \(y = -5\) (horizontal) → perpendicular.
Find the point of intersection of \(x = 7\) and \(y = -9\).
Answer: \((7, -9)\)
Rewrite \(|y| = |x|\) as two straight-line equations.
Because \(|y| = |x|\), either \(y = x\) or \(y = -x\).
Answer: \(y = x\) and \(y = -x\)
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\). It measures the vertical change divided by the horizontal change between two points.
An undefined slope occurs for a vertical line, where the change in \(x\) is zero. Division by zero is undefined.
Rewrite the equation as \(y = mx + b\); the coefficient of \(x\) is the slope. Alternatively, for \(Ax + By = C\), the slope is \(-A/B\).
Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: \(m_1 m_2 = -1\).
Slope-intercept form is \(y = mx + b\). Standard form is \(Ax + By = C\), usually with \(A > 0\) and \(A, B, C\) integers.