Problem 21 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Slope \(=\frac{1}{2},\) passing through the origin

Step-by-Step Solution

Answer

The line has the same equation in both point-slope form and slope-intercept form, which is \(y = 1/2 x\).

Step 1: Formulate the Point-Slope Form

The point-slope form of the equation of a line is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) are the coordinates of a point on the line. Here, the slope \(m = 1/2\) and since the line passes through the origin, the point is the origin \((0, 0)\). Substituting these values into the equation, it simplifies to: \(y - 0 = 1/2 (x - 0)\) or simply \(y = 1/2 x\).

Step 2: Formulate the Slope-Intercept Form

The slope-intercept form of the equation of a line is given by \(y = mx + c\), where \(m\) is the slope and c is the y-intercept. Given that the slope \(m = 1/2\) and since the line passes through the origin, the y-intercept is 0. Substituting these values into the equation, it simplifies to: \(y = 1/2x + 0\), which is simply \(y = 1/2x\).

Key Concepts

Point-Slope Form

The point-slope form is a handy way to write the equation of a line when you know a point on the line and the slope. It uses the formula:

• \(y - y_1 = m(x - x_1)\)

where \(m\) is the slope, and \(x_1, y_1)\) are the coordinates of a specific point on the line. This form is particularly useful because it directly uses a given point and slope to establish the line equation. You substitute \(m\) with the line's slope and use the coordinates of the known point to plug into \(x_1, y_1)\).

In our exercise, the slope \(m\) is \(\frac{1}{2}\) and the point is the origin, \((0, 0)\). Plugging these into the point-slope formula gives:

• \(y - 0 = \frac{1}{2}(x - 0)\)
• This simplifies to: \(y = \frac{1}{2}x\)

This simplified form makes it easier to identify or write more complex situations where you might know varied points or different slopes.

Slope-Intercept Form

The slope-intercept form is another powerful equation format for a line. This format is about clarity and easy visualization of the slope and y-intercept. It is expressed as:

• \(y = mx + c\)

where \(m\) represents the slope of the line and \(c\) the y-intercept. The advantages of this form include the straightforward way it displays how the line progresses in terms of rise over run and where it crosses the y-axis.

In this exercise, we know that the slope \(m\) is \(\frac{1}{2}\) and the line passes through the origin. This means the y-intercept \(c\) is 0. Substituting these values gives:

• \(y = \frac{1}{2}x + 0\)
• Simply, it is \(y = \frac{1}{2}x\).

This represents a line that increases moderately, rising one unit for every two units it moves to the right, always crossing through the origin.

Slope

The slope is an essential characteristic of a line and a key component in both point-slope and slope-intercept forms. It is essentially the measure of how steep or flat the line is.

The slope \(m\) is calculated as the 'rise' over the 'run', or the change in the y-coordinates divided by the change in the x-coordinates. This can be expressed as:

• \(m = \frac{\Delta y}{\Delta x}\)
• Alternatively, if given two points, \((x_1, y_1)\) and \(x_2, y_2)\), the slope can be calculated with: \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

A positive slope means the line rises as it moves from left to right, while a negative slope indicates it falls. A slope of zero means the line is horizontal, and an undefined slope (division by zero) means the line is vertical.

For our line with a slope of \(\frac{1}{2}\), it signifies that for every two units moved horizontally, the line rises by one unit vertically. This consistent pattern is crucial for graphing and understanding linear motion and relationships.