Problem 22 - College Algebra

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 line has the following forms: Point-slope: \(y = -\frac{1}{3}x\) and Slope-intercept: \(y = -\frac{1}{3}x\).

Step 1: Identify the given slope and point

The slope \(m = -\frac{1}{3}\) and the line passes through the origin (0,0).

Step 2: Write the point-slope form of the line

The general point-slope form of a line is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) are the co-ordinates of a point on the line and \(m\) is the slope. Substituting the given slope and point into the formula, we get \(y - 0 = -\frac{1}{3}(x - 0)\), or simply \(y = -\frac{1}{3}x\).

Step 3: Write the slope-intercept form of the line

The general slope-intercept form of a line is \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. We know that the y-intercept is 0, because the line passes through the origin. Therefore, the slope-intercept form of this line is also \(y = -\frac{1}{3}x + 0\), or simply \(y = -\frac{1}{3}x\).

Key Concepts

Point-Slope Form

The point-slope form of a linear equation is a handy tool when you know the slope of a line and a point the line passes through. It is expressed as: \[ y - y_1 = m(x - x_1) \] In this equation,

• \( m \) represents the slope of the line.
• \((x_1, y_1)\) are the coordinates of the given point.

This form is particularly useful because it gives you the equation of the line using just these two pieces of information.
In the exercise, we have a slope \(-\frac{1}{3}\) and a point that is the origin \((0,0)\). By substituting these into the point-slope formula, the equation becomes: \[ y - 0 = -\frac{1}{3}(x - 0) \] which simplifies directly to \( y = -\frac{1}{3}x \), since subtracting zero doesn't change the terms. This equation tells us how the line declines as you move along the x-axis.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to express a linear equation. This form highlights both the slope of the line and the y-intercept. It is represented by the formula: \[ y = mx + b \] Here:

• \( m \) is the slope of the line, indicating the line's steepness.
• \( b \) is the y-intercept, the point where the line crosses the y-axis.

The beauty of this form is that it is clear and easy to graph.
The exercise uses the slope \(-\frac{1}{3}\) and the fact that the line passes through the origin \((0,0)\). Since the line goes through the origin, the y-intercept is 0. So, plugging these values into the slope-intercept form, we have: \[ y = -\frac{1}{3}x + 0 \] which simplifies to \( y = -\frac{1}{3}x \). Both forms, point-slope and slope-intercept, define the same line, just emphasized differently.

Slope Through Origin

A line that passes through the origin \((0,0)\) is unique because its y-intercept \( b \) is always 0. In both point-slope and slope-intercept forms, this simplifies the equation significantly.
When a line passes through the origin, you don't have to worry about calculating \( b \), which simplifies your calculations.

• In point-slope form, let's recall: \( y - y_1 = m(x - x_1) \). Here, because \((x_1, y_1)\) is \((0,0)\), the equation reduces to \( y = mx \).
• In slope-intercept form \( y = mx + b \), since \( b = 0 \), this also reduces to \( y = mx \).

Thus, a line through the origin is straightforward: its equation will always be in the form of \( y = mx \), showing directly how every point on the line relates linearly through the slope \( m \).