Problem 86 - 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 \((-5,4) \quad x+y=8\)

Step-by-Step Solution

Answer

The equation of the line parallel to the given line and passing through the point (-5, 4) is \(y = -x - 1\). The equation of the line perpendicular to the given line and passing through the point (-5, 4) is \(y = x + 9\).

Step 1: Find the slope of the given line

Rewrite the given line \(x + y = 8\) in slope-intercept form \(y = mx + b\) where \(m\) is the slope. After rearranging, the line becomes \(y = -x + 8\). So, the slope of the given line is \(-1\).

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

Since parallel lines have the same slope, the line parallel to the given line that passes through the point (-5, 4) will also have a slope of -1. Using the point-slope form of a line equation, \(y - y1 = m(x - x1)\), where \(m\) is the slope and \((x1, y1)\) is the given point, substitute the values to get the equation \(y - 4 = -1(x + 5)\). After simplifying, the equation of the line parallel to the given line is \(y = -x - 1\).

Step 3: Write the equation of the line perpendicular to the given line

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. Therefore, the line perpendicular to the given line will have a slope of \(1\). Again, using the point-slope form, substitute to get \(y - 4 = 1(x + 5)\). Hence, the equation of the line perpendicular to the given line is \(y = x + 9\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a very common way to express the equation of a line. It is written as \[y = mx + b\]where:

• m is the slope of the line, indicating its steepness and direction.
• b is the y-intercept, the point where the line crosses the y-axis.

This format is especially useful because it quickly shows both the slope and y-intercept, making it easy to graph the line. To convert an equation to this form, ensure 'y' is isolated on one side of the equation. For example, the equation \(x + y = 8\) can be rearranged to become \(y = -x + 8\). Here, the slope \(m\) is -1, and the y-intercept \(b\) is 8. Thus, a line with the equation \(y = -x + 8\) travels downward from left to right.

Point-Slope Form

Point-slope form is ideal for writing the equation of a line when you know a point on the line and its slope. It is expressed as:\[y - y_1 = m(x - x_1)\]In this equation:

• (x₁, y₁) represents a specific point on the line.
• m is the slope.

This form is handy in scenarios where you either don't immediately have the y-intercept or are working with a specific point. By plugging in the slope you have and the coordinates of the given point, it is straightforward to derive the specific linear equation. For instance, using the scenario where the point is (-5, 4) with a slope of -1, the point-slope form becomes:\[y - 4 = -1(x + 5)\]This can lead to further simplifications and transformations, making it a flexible choice for constructing linear equations from basic data.

Parallel Lines

Parallel lines, in simple terms, are lines that run alongside each other and never meet. They have identical slopes, which means they ascend or descend at the same rate:

• If a line has a slope of \(m\), any line parallel to it will also have a slope of \(m\).

Using the slope-intercept form, if you know a line's equation, finding a parallel line that passes through a specific point is a breeze. From the exercise, if the line is represented as \(y = -x + 8\), its parallel counterpart that cuts through (-5, 4) will similarly maintain the slope of -1. Using the point-slope form and knowing the parallel lines share the same slope, you get the equation:\[y - 4 = -1(x + 5)\]This simplifies to the parallel line in question: \(y = -x - 1\). The line remains parallel by adhering to the original line's gradient while adjusting around a new point.

Perpendicular Lines

Perpendicular lines make a 90-degree angle with one another and exhibit a special slope relationship. To form perpendicular lines, the slope of one line must be the negative reciprocal of the other:

• If one line has a slope of \(m\), the perpendicular line will have a slope of \(-\frac{1}{m}\).

This feature makes slope calculations more exciting, as flipping the fraction and changing the sign sets the stage for perpendicular intersection. In cases of vertical and horizontal lines, the slopes shift from undefined or 0 to create this perpendicularity.Returning to the exercise where the line's slope is -1, the perpendicular line must have a slope of 1 (since the negative reciprocal of -1 is 1). Hence, the equation for a perpendicular line passing through (-5, 4) takes shape as:\[y - 4 = 1(x + 5)\]After simplifying, you find the perpendicular line equation: \(y = x + 9\). Here, this alteration in slope direction ensures the lines meet at right angles, offering a classically satisfying visual cue.