Problem 87 - College Algebra and Calculus: An Applied Approach

Question

Write equations of the lines through the point (a) parallel to the given line and (b) perpendicular to the given line. Point \(\quad\) Line \(\begin{array}{ll}\underline{\phantom{xxx}}\left(\frac{1}{4},-\frac{2}{3}\right) & 2 x-3 y=5\end{array}\)

Step-by-Step Solution

Answer

The line parallel to the given line through \((\frac{1}{4}, -\frac{2}{3})\) is \(y = \frac{2}{3}x - \frac{3}{4}\) and the line perpendicular to the given line through \((\frac{1}{4}, -\frac{2}{3})\) is \(y = -\frac{3}{2}x + \frac{1}{6}\).

Step 1: Find the slope of the given line

First, the equation of the given line should be converted to slope-intercept form. This is achieved by rewriting \(2x-3y=5\) to \(y = \frac{2}{3}x - \frac{5}{3}\). From this form, the slope \(m_1\) of the given line can be identified as \(\frac{2}{3}\).

Step 2: Write equation of line parallel to the given line

The slope of a line parallel to the given line will be equal to the slope of the given line. So, the slope \(m_2\) of the new line is also \(\frac{2}{3}\). Using the point-slope form of a line \(y-y_1=m(x-x_1)\), where \((x_1,y_1)=(\frac{1}{4} , -\frac{2}{3})\) is the given point, we get the equation of the line as \(y + \frac{2}{3}=\frac{2}{3}(x - \frac{1}{4})\) which simplifies to \(y = \frac{2}{3}x - \frac{3}{4}\).

Step 3: Find the slope of line perpendicular to the given line

The slope of a line perpendicular to the given line will be the negative reciprocal of the slope of the given line. So, the new slope \(m_3\) is -\frac{3}{2}.

Step 4: Write equation of line perpendicular to the given line

Using the point-slope form of a line again with \((x_1,y_1) = (\frac{1}{4} , -\frac{2}{3})\) and \(m_3 = -\frac{3}{2}\), the equation of the line is given by \(y + \frac{2}{3} = -\frac{3}{2}(x - \frac{1}{4})\), which simplifies to \(y = -\frac{3}{2}x + \frac{1}{6}\).

Key Concepts

Parallel and perpendicular lines

Understanding parallel and perpendicular lines is crucial in geometry and algebra. When two lines are parallel, they will never intersect and they have the identical slope. This means their slopes are equal. For example, if a line has a slope of \(\frac{2}{3}\), a parallel line will also have a slope of \(\frac{2}{3}\).

On the other hand, perpendicular lines intersect at a right angle (90 degrees). The slopes of these lines are negative reciprocals of each other. If one line has a slope of \(\frac{2}{3}\), a line perpendicular to it will have a slope of \(-\frac{3}{2}\). Why a negative reciprocal? This relationship ensures that the lines cross at a perfect 90-degree angle. Note that you can find a negative reciprocal by flipping a fraction and changing its sign.

• Parallel lines: same slope
• Perpendicular lines: negative reciprocal slopes

Slope-intercept form

The slope-intercept form is a common way to represent a linear equation. It is expressed as \( y = mx + b \) where:

• \( m \) is the slope of the line
• \( b \) is the y-intercept, the point where the line crosses the y-axis

This form is particularly useful for quickly identifying the slope and y-intercept of a line. By looking at the equation \(y = \frac{2}{3}x - \frac{5}{3}\), you can instantly determine that the line has a slope of \(\frac{2}{3}\) and crosses the y-axis at \(-\frac{5}{3}\).

To convert an equation to this form, isolate \(y\) on one side of the equation. For example, solving \(2x - 3y = 5\) for \(y\) gives \(y = \frac{2}{3}x - \frac{5}{3}\). This transformation puts the equation neatly into slope-intercept form.

Point-slope form

The point-slope form of a linear equation is very handy when you know a point on the line and the slope. It is given by the formula:

\[ y - y_1 = m(x - x_1) \]

Where:

• \((x_1, y_1)\) is a known point on the line
• \(m\) is the slope

This form directly uses the slope and a specific point to create the equation of a line, making it very practical for quickly writing equations. For example, if a line passes through the point \((\frac{1}{4}, -\frac{2}{3})\) and has a slope of \(\frac{2}{3}\), the point-slope form equation is:

\[ y + \frac{2}{3} = \frac{2}{3}(x - \frac{1}{4}) \]

By simplifying this, you can transition to the slope-intercept form if needed, which in this case gives \(y = \frac{2}{3}x - \frac{3}{4}\). It’s a useful equation form that connects directly to real points and slopes.