Problem 22 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=\frac{1}{3},\) passing through the origin

Step-by-Step Solution

Answer

The equation of the line in point-slope form is \(y = \frac{1}{3}x\), and in slope-intercept form, it's \(y = \frac{1}{3}x\).

Step 1: Write the Point-Slope Form

Substitute the given point and the slope into the point-slope form equation: \(y - y_1 = m(x - x_1)\). Since the given point is the origin (0,0), it simplifies it to \(y = \frac{1}{3}x\).

Step 2: Write the Slope-Intercept Form

We are given that the line passes through the origin, which means the y-intercept is 0. Substitute the slope and the y-intercept into the slope-intercept form equation \(y = mx + b\). This gives the slope-intercept form as \(y = \frac{1}{3}x + 0\), which simplifies to \(y = \frac{1}{3}x\). Thus, the equations in both forms are the same in this case.

Key Concepts

slope-intercept form

The slope-intercept form is one of the most common ways to represent the equation of a line. It is expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line. The slope indicates how steep the line is and in which direction it inclines.
• \(b\) is the y-intercept. This is the point where the line crosses the y-axis.

In our exercise, the line passes through the origin (0,0), so the y-intercept \(b\) is 0. Therefore, the slope-intercept form can be written as \(y = \frac{1}{3}x\).
This equation means that for every unit increase in \(x\), \(y\) increases by \(\frac{1}{3}\). This straightforward format is particularly helpful for quickly identifying how a line behaves.

slope

The slope of a line measures its steepness and is usually denoted by \(m\). It's calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. In mathematical terms, the slope formula is \( m = \frac{y_2-y_1}{x_2-x_1} \).

• A positive slope means the line is rising from left to right.
• A negative slope indicates the line is falling from left to right.
• A zero slope represents a horizontal line, and an undefined slope corresponds to a vertical line.

In the provided problem, the slope is \(\frac{1}{3}\). This means that for every increase of 3 units in the x-direction, the line rises by 1 unit in the y-direction. Understanding the slope helps to visualize and construct the line quickly.

equation of a line

The equation of a line can be written in various forms, but they all describe the same concept: the set of all points that lie along the line. The general forms include:

• Point-Slope Form: \(y - y_1 = m(x - x_1)\)
• Slope-Intercept Form: \(y = mx + b\)
• Standard Form: \(Ax + By = C\)

In this exercise, we used both the point-slope and slope-intercept forms. The point-slope form is particularly useful when you know a point on the line and the slope. With a point at the origin (0,0) and a slope of \(\frac{1}{3}\), the equation becomes \( y = \frac{1}{3}x \). The slope-intercept form is simple as well with the known values, leading to the same equation, \( y = \frac{1}{3}x \). Knowing these forms allows you to effectively craft an equation based on different starting information.