Question

Write an equation for the line through \((3,-3)\) that is (a) parallel to the line \(y=2 x+5\); (b) perpendicular to the line \(y=2 x+5\); (c) parallel to the line \(2 x+3 y=6\); (d) perpendicular to the line \(2 x+3 y=6\); (e) parallel to the line through \((-1,2)\) and \((3,-1)\); (f) parallel to the line \(x=8\); (g) perpendicular to the line \(x=8\).

Short answer

(a) \(y=2x-9\), (b) \(y=-\frac{1}{2}x-\frac{3}{2}\), (c) \(y=-\frac{2}{3}x-1\), (d) \(y=\frac{3}{2}x-\frac{15}{2}\), (e) \(y=-\frac{3}{4}x-\frac{3}{4}\), (f) \(x=3\), (g) \(y=-3\).

Step-by-step solution

Understand Parallel Lines

A line parallel to another line shares the same slope. Given the line equation in slope-intercept form \(y = mx + b\), the coefficient \(m\) represents the slope.

Slope of Given Line for (a) and (b)

For the line \(y = 2x + 5\), the slope \(m\) is 2. This is the parallel slope for part (a) and the slope for the perpendicular will be the negative reciprocal.

Equation for a Line Parallel to \(y=2x+5\)

Using the point-slope form \(y - y_1 = m(x - x_1)\) and point \((3,-3)\), the equation is \(y + 3 = 2(x - 3)\). Simplify it to \(y = 2x - 9\).

Equation for a Line Perpendicular to \(y=2x+5\)

The negative reciprocal of slope 2 is \(-\frac{1}{2}\). Using point \((3,-3)\), the equation is \(y + 3 = -\frac{1}{2}(x - 3)\) which simplifies to \(y = -\frac{1}{2}x - \frac{3}{2}\).

Rewrite Equation for (c) and (d) in Slope-Intercept Form

Given line \(2x + 3y = 6\). Rearrange to get \(y = -\frac{2}{3}x + 2\). The slope is \(-\frac{2}{3}\).

Equation Parallel to \(2x+3y=6\)

Using slope \(-\frac{2}{3}\) and point \((3,-3)\), use the point-slope form: \(y + 3 = -\frac{2}{3}(x - 3)\). Simplify to \(y = -\frac{2}{3}x - 1\).

Equation Perpendicular to \(2x+3y=6\)

The negative reciprocal of \(-\frac{2}{3}\) is \(\frac{3}{2}\). With point \((3,-3)\), use \(y + 3 = \frac{3}{2}(x - 3)\). Simplify to \(y = \frac{3}{2}x - \frac{15}{2}\).

Find Slope of the Line Through Points \((-1,2)\) and \((3,-1)\)

Calculate slope as \(m = \frac{-1 - 2}{3 - (-1)} = -\frac{3}{4}\). This slope is used in part (e).

Equation Parallel to Line Through Two Points

For slope \(-\frac{3}{4}\) and point \((3,-3)\), use \(y + 3 = -\frac{3}{4}(x - 3)\). Simplify to \(y = -\frac{3}{4}x - \frac{3}{4}\).

Equation Parallel to Vertical Line \(x=8\)

A line parallel to \(x=8\) is also vertical. Since it passes through \((3,-3)\), the equation is \(x = 3\).

Equation Perpendicular to Vertical Line \(x=8\)

A line perpendicular to a vertical line is horizontal. Through point \((3,-3)\), it is \(y = -3\).

Key concepts

Parallel Lines

Parallel lines are lines in a plane that are always the same distance apart and never touch. These lines have the same slope, which is a number that describes how steep the line is. A parallel line shares its slope with the line it is parallel to.

• If you know one line's equation is in the form of slope-intercept, like \( y = mx + b \), the slope \( m \) tells you how to find another line parallel to it. For example, the line \( y = 2x + 5 \) has a slope of 2. Any line with a slope of 2 will be parallel to it.
• Finding a parallel line involves using the same slope and plugging it into the point-slope form if a specific point on the line is given. For instance, to find a line parallel to \( y = 2x + 5 \) and passing through \((3, -3)\), use the slope 2 and the point \( (3, -3) \).
• Vertical lines, like \( x = 8 \), are unique because they never use the typical \( y = mx + b \) form. Instead, any line that is vertical, like \( x = 3 \), will be parallel. Such lines remain vertical, sharing the same direction and never meeting.

Perpendicular Lines

Perpendicular lines are lines that intersect at a right angle (90 degrees). If two lines are perpendicular, their slopes are negative reciprocals. This means if one line's slope is \( m \), the other line's slope will be \(-\frac{1}{m}\).

• For example, a line with a slope of 2 has a perpendicular slope of \(-\frac{1}{2}\). So if you're asked to find a line perpendicular to \( y = 2x + 5 \), you should use \(-\frac{1}{2}\) as the slope.
• With a point like \((3, -3)\) given, the point-slope form is used to find the perpendicular line: \( y + 3 = -\frac{1}{2}(x - 3) \) simplifies to \( y = -\frac{1}{2}x - \frac{3}{2} \).
• Horizontal and vertical lines are perfect examples of perpendicular relationships. A vertical line, such as \( x = 8 \), and a horizontal line, like \( y = -3 \), are perpendicular because vertical and horizontal lines always meet at right angles.

Slope-Intercept Form

The slope-intercept form is a common way to write a linear equation. It is written as \( y = mx + b \), where \( m \) represents the slope (how steep the line is) and \( b \) is the y-intercept (where the line crosses the y-axis). Simple to use, this form helps identify important features of the line right away.

• When you look at an equation in slope-intercept form, you see directly what the slope is, which makes finding parallel or perpendicular lines easier.
• For instance, the equation \( y = 2x + 5 \) shows right away that the slope is 2 and the y-intercept is 5. This is useful to know when solving problems related to parallel or perpendicular lines.
• To convert an equation into slope-intercept form, you might need to rearrange the terms. For example, transforming \( 2x + 3y = 6 \) into \( y = -\frac{2}{3}x + 2 \) makes it clearer what the slope and y-intercept are. That way, it's ready for analyzing line relationships.

Point-Slope Form

The point-slope form is handy when you know a point on the line and the line’s slope. The equation is written as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a known point and \( m \) is the slope. This form makes it easy to craft the equation of a line when these two details are known.

• This is especially useful for writing equations for lines that are either parallel or perpendicular to another line, as you frequently start with a known point and a known slope.
• Consider you have a point \((3, -3)\) and want a line parallel to \( y = 2x + 5 \). The slope is 2, so the point-slope form becomes \( y + 3 = 2(x - 3) \), simplifying to \( y = 2x - 9 \).
• Learning to switch between point-slope form and slope-intercept form can help solve a variety of problems easily, always allowing clear insights into the line's behavior and its direction.