Problem 58 - College Algebra and Calculus: An Applied Approach

Question

Find an equation of the line passing through the points. \((1,1),\left(6,-\frac{2}{3}\right)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points (1,1) and \( \left(6,-\frac{2}{3}\right) \) is \( y = -\frac{1}{3}x + \frac{4}{3} \)

Step 1: Calculating the Slope

The slope (m) of a line passing through the points (x1, y1) and (x2, y2) is given by the formula: m = (y2 - y1) / (x2 - x1). Substituting (x1, y1) = (1,1) and (x2, y2) = \( \left(6,-\frac{2}{3}\right) \), we get: m = \( \left(-\frac{2}{3} - 1\right) / (6 - 1) = -\frac{5}{3} / 5 = -\frac{1}{3} \)

Step 2: Using the Slope-Intercept Form

The equation of a line in slope-intercept form is y = mx + b. Since from Step 1 we know that m = -1/3, we can replace m in the equation, getting y = -1/3*x + b. To find the y-intercept (b), we substitute one of the given points into the equation. Let's use point (1,1), obtaining 1 = -1/3*1 + b, which gives b = 1 + 1/3 = 4/3.

Step 3: Writing the Equation of the Line

Finally, we substitute m and b from Steps 1 and 2 back into the general linear equation y = mx + b, yielding the equation of the line: y = -1/3*x + 4/3

Key Concepts

Slope Formula

To find the slope of a line that passes through two points, we use the slope formula. Slope is a measure of steepness and is essential to defining a line. The formula for slope \( m \) is given by:

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

Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Essentially, the formula calculates the change in \( y \) (vertical change) over the change in \( x \) (horizontal change).
Think of it as the rise over run. In the given problem, substituting the points \((1, 1)\) and \( \left(6, -\frac{2}{3}\right) \) into the formula, we calculate the slope as follows:

• \( m = \frac{-\frac{2}{3} - 1}{6 - 1} = \frac{-\frac{5}{3}}{5} = -\frac{1}{3} \)

This slope of \(-\frac{1}{3}\) tells us the line falls one unit for every three units it moves to the right.

Slope-Intercept Form

The slope-intercept form is a commonly used format for linear equations that allows easy access to the slope and y-intercept of a line. It is written as:

• \( y = mx + b \)

Where \( m \) is the slope and \( b \) is the y-intercept. This form is useful because it provides a clear understanding of the line's behavior at a glance.
To use the slope-intercept form, you need the line’s slope and one point on the line to find \( b \).
Using the equation \( y = -\frac{1}{3}x + b \), we substitute the point \((1,1)\):

• \( 1 = -\frac{1}{3}(1) + b \)
• Simplifying gives: \( 1 = -\frac{1}{3} + b \)
• \( b = 1 + \frac{1}{3} = \frac{4}{3} \)

Thus, the y-intercept \( b \) is \( \frac{4}{3} \), meaning the line crosses the y-axis at this point.

Linear Equation

A linear equation is an algebraic equation describing a straight line. It involves variables that do not have exponents other than one, making it a first-degree polynomial.
Linear equations can be expressed in various forms, but the slope-intercept form \( y = mx + b \) is often easiest for graphing and analysis.
For a line passing through \((1, 1)\) with a slope of \(-\frac{1}{3}\), and y-intercept \( \frac{4}{3} \), the equation is:

• \( y = -\frac{1}{3}x + \frac{4}{3} \)

This equation shows how \( y \) depends on \( x \), with changes in \( x \) affecting \( y \) according to the slope. Adding the y-intercept adjusts the entire line's position up or down on the graph. Understanding linear equations is crucial not only in mathematics but also in real-world applications like physics and economics, as they model relationships that have a constant rate of change.