Problem 21 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The equation of the line both in point-slope form and slope-intercept form is \(y = \frac{1}{2}x\).

Step 1: Deriving the point-slope form equation

First, we need to write the equation of the line using the point-slope form \(y - y1 = m(x - x1)\). In this formula, plug in the slope which is \(\frac{1}{2}\) for the \(m\) and the point \(0,0\) for \((x1, y1)\).So, the equation becomes \(y - 0 = \frac{1}{2} (x - 0)\) which simplifies to \(y = \frac{1}{2}x\).

Step 2: Deriving the slope-intercept form equation

Now, we need to write the equation of the line using the slope-intercept form \(y = mx + b\). In this formula, plug in the slope \(\frac{1}{2}\) for \(m\) and since it passes through the origin, \(b\) will be \(0\). So, the equation becomes \(y = \frac{1}{2}x + 0\), which simplifies to \(y = \frac{1}{2}x\).

Key Concepts

slope-intercept form

The slope-intercept form of a linear equation is one of the most common ways to represent a line graphically. This form is given by the equation \(y = mx + b\), where:

• \(m\) represents the slope of the line, showing how steep it is.
• \(b\) represents the y-intercept, which is the point where the line crosses the y-axis.

This form is especially useful because it provides immediate information about the slope and the y-intercept of the line.
In this particular exercise, the slope \(m\) is given as \(\frac{1}{2}\), and since the line passes through the origin (0,0), the y-intercept \(b\) is 0. Therefore, the slope-intercept form is simplified to \(y = \frac{1}{2}x\). This simplicity makes it easy to graph and understand the relationship of the line to the coordinate plane.

linear equations

Linear equations are expressions that create straight lines when graphed on a coordinate plane. They are written in the form \(ax + by = c\), but can also be rearranged into different formats such as point-slope form and slope-intercept form.
Linearity implies a constant rate of change, signified by the slope. These equations can also tell us whether lines are parallel, intersecting, or overlapping by comparing their slopes.

• Lines with equal slopes are parallel and don’t intersect.
• Lines with different slopes intersect at exactly one point.

In our specific exercise, the linear equation \(y = \frac{1}{2}x\) shows a direct relationship between the x and y variables. This simple linear relationship tells us that for every 2 units moved horizontally, there is 1 unit of vertical movement, as defined by the slope of \(\frac{1}{2}\).

slope of a line

The slope of a line is a measure of its steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, slope \(m\) is calculated as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
In our exercise, the slope is given as \(\frac{1}{2}\). This indicates a gentle incline of the line, where for every 2 units moved horizontally along the x-axis (run), there is a rise of 1 unit vertically along the y-axis. The slope tells us not only how steep the line is but also its direction:

• A positive slope means the line inclines upwards as it moves from left to right.
• A negative slope indicates the line decreases or falls as you move from left to right.

The given slope of \(\frac{1}{2}\) for the line through the origin perfectly demonstrates a positive, though not steep, incline.