Problem 21 - Algebra for College Students

Question

Determine the linear function whose graph is a line that is perpendicular to the line \(g(x)=5 x-2\) and contains the point \((6,3) . \quad f(x)=-\frac{1}{5} x+\frac{21}{5}\)

Step-by-Step Solution

Answer

The linear function is \( f(x) = -\frac{1}{5}x + \frac{21}{5} \).

Step 1: Identify the Slope of the Given Line

The given line is \( g(x) = 5x - 2 \). The slope of this line is 5, which is the coefficient of \( x \).

Step 2: Find the Perpendicular Slope

Two lines are perpendicular if the product of their slopes is -1. Therefore, if one line has a slope of 5, the perpendicular slope, \( m \), satisfies \( 5m = -1 \). Solving for \( m \) gives \( m = -\frac{1}{5} \).

Step 3: Use the Point-Slope Form of a Line

To find the equation of a line, use the point-slope form equation: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. Here, the point is \( (6, 3) \) and \( m = -\frac{1}{5} \). Thus, \( y - 3 = -\frac{1}{5}(x - 6) \).

Step 4: Simplify the Equation

Distribute the slope on the right side: \( y - 3 = -\frac{1}{5}x + \frac{6}{5} \). Then, add 3 to both sides to isolate \( y \): \( y = -\frac{1}{5}x + \frac{6}{5} + 3 \). Convert 3 to a fraction with 5 as a denominator: \( 3 = \frac{15}{5} \). So, the equation becomes \( y = -\frac{1}{5}x + \frac{21}{5} \).

Key Concepts

Linear Functions

A linear function is one of the simplest types of functions used in algebra. It describes a line on the graph and has the general form:

• \( y = mx + b \)

Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis.
The slope \( m \) indicates how steep the line is, and the y-intercept \( b \) tells us the starting point of the line when \( x = 0 \).
Linear functions are extremely useful for modeling relationships between variables that change at a constant rate. They create straight lines on a coordinate plane. The simplicity of linear functions makes them foundational in mathematics and a stepping stone to understanding more complex functions.

Point-Slope Form

The point-slope form is another way to express the equation of a line. It is particularly helpful when you know a single point on the line and its slope. The point-slope form is written as:

• \( y - y_1 = m(x - x_1) \)

In this formula, \((x_1, y_1)\) is the known point on the line, and \( m \) is the line's slope.
This form allows you to rapidly set up an equation for a straight line when it's easier to start with a point and a slope rather than finding the y-intercept directly.
Using point-slope form can simplify the process of finding the equation of a line, especially when you are given specific conditions, like a perpendicular line through a specific point.

Slope of a Line

The slope of a line is a measure of its steepness and direction. To find the slope between any two points on a line, you can use the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line.
The slope tells us how much \( y \) changes for a given change in \( x \).
In simpler terms, the slope describes if the line is rising or falling as it moves from left to right across the graph.
A positive slope means the line ascends, whereas a negative slope means it descends. The slope is essential for defining the direction and angle of a line, especially when working with perpendicular or parallel lines.