Lines

Lines, parallel, perpendicular, slope intercept, point-slope.

Introduction

As was mentioned in the last section, lines are one of the geometric object that requires the least amount of information to distinguish them. The only information we need is two points. From this information we can find the slope of the line. From here we can determine if lines are two lines are parallel or perpendicular. Then there are two common ways to write the equation of a line. These are called the slope intercept form and the point slope form.

Slope, Parallel, Perpendicular

As was mentioned before the first thing we will discuss is the slope of a line. Given two points, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1) \text{ and }(x_2, y_2)} the formula for the slope is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{y_2 - y_1}{x_2 - x_1}} . From a geometric standpoint the slope tells you how quickly the line increases in y-value per each unit change in x-value.

Example: Find the slope of the line containing (5, 3) and (6, 9).

The slope, denoted m, is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{9-3}{6-5} = 6 = \frac{3-9}{5-6}} .

Two lines Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = m_1 x + b_1 \text{ and } y = m_2 x + b_2} are parallel if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1 = m_2}

For example, y = 5x + 3 is parallel to y = 5x - 4, but is not parallel to y = -3x + 8.

Two lines, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = m_1x +b_1 \text{ and } y = m_2x + b_2} are perpedicular if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1 = \frac{-1}{m_2}} .

For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 3x - 2} is perpendicular to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \frac{-1}{3}x + 5} , but is not parallel to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = 3x + 5}

Slope intercept Form

The slope intercept form for a line provides us with both the slope, and y intercept without requiring any work. It also allows us to write the equation of a line with only two pieces of information, the slope and y-intercept. Given a slope m, and y-intercept (0, b) the slope intercept form for the line is y = mx + b.

Example: Given a line with slope 5 and y-intercept (0, -2) the equation of the line, in slope intercept form, is y = 5x - 2.

Point Slope Form

The point slope form is a more widely applicable way to write the equation of a line. Just like the slope intercept form we only require knowledge of the slope, but we only need a point that is on the line, even if it is not the y-intercept. Given a point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1)} and a slope m, the point slope form for a line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - y_1 = m(x - x_1)} .

Example: Find the equation of a line with slope -2 going through the point (3, 7).

Solution: The equation for the line is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y - 7 = (-2)(x - 3)}

Return to Topics Page