Problem 22 - College Algebra

Question

Write an equation in slope-intercept form of a linear function \(f\) whose graph satisfies the given conditions. The graph of \(f\) passes through \((-5,6)\) and is perpendicular to the line that has an \(x\) -intercept of \(3\) and a \(y\) -intercept of \(-9\)

Step-by-Step Solution

Answer

The equation in slope-intercept form of the line that satisfies the provided conditions is \(y=\frac{1}{3}x+\frac{23}{3}\)

Step 1: Find the slope of the original line

Given the x-intercept (3) and y-intercept (-9) for the original line, remember the formula for the slope of a line when given two points \((x1, y1)\) and \((x2, y2)\) is \(m = \frac{y2 - y1}{x2 - x1}\), in this case, \(m = \frac{-9 - 0}{3 - 0} = -3\)

Step 2: Find Slope for Perpendicular Line

The slope of a line perpendicular to a line with slope \(m_o\) is given by: \(m_p = -\frac{1}{m_o}\). Hence, the second slope is \(m_p = -\frac{1}{-3} = \frac{1}{3}\)

Step 3: Write Equation in Slope-Intercept Form

Applying point-slope form: \(y - y_1 = m(x - x_1)\), where \(m = \frac{1}{3}\) and using point (-5,6) , results in \(y=\frac{1}{3}(x+5) +6 = \frac{1}{3}x +\frac{5}{3}+6 = \frac{1}{3}x+\frac{23}{3}\). Our slope-intercept equation is therefore \(y=\frac{1}{3}x+\frac{23}{3}\)

Key Concepts

Linear Function

A linear function is one of the most fundamental concepts in algebra. It represents a straight line when graphed on a Cartesian plane. The general form of a linear equation is written as
\( y = mx + b \),
where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses the y-axis. The slope \( m \) indicates how steep the line is, with positive values meaning the line goes upwards, and negative values signaling a downward trend as you move from left to right.

The simplicity of linear functions makes them immensely useful in representing real-world situations where there is a constant rate of change, like the relationship between time and distance in uniformly accelerated motion, or cost and quantity in economics.

Perpendicular Lines Slope

Understanding the relationship between slopes of perpendicular lines is key in coordinate geometry. Two lines are perpendicular if they intersect to form a right angle. The slopes of these two lines have a special relationship: they are negative reciprocals of each other. This means if one line has a slope of \( m \), the line perpendicular to it will have a slope of \( -\frac{1}{m} \).

For example, if a line has a slope of 2, a line perpendicular to it will have a slope of \( -\frac{1}{2} \). This relationship is essential when finding the equation of a line that must meet a specific condition, like being perpendicular to another line at a given point, as seen in the exercise solution. Such knowledge is not just academically relevant; it's fundamental for fields like construction and navigation, where precise angles matter.

Point-Slope Form

The point-slope form of a line is another way to write the equation of a line. It is particularly handy when we know a specific point through which the line passes and its slope. The formula is given by
\( y - y_1 = m(x - x_1) \),
where \( (x_1, y_1) \) is the particular point on the line, and \( m \) is the slope. This form can be manipulated to find the slope-intercept form or to start creating equations based on given geometrical conditions.

In practical terms, point-slope form can quickly help us draft the behavior of a line – maybe in science for a prediction of a trend based on data points or in architecture when determining the pitch of a roof. In many situations, it's about finding the line that tells the story of the data or the requirements of a physical structure.