Problem 38 - Applied Calculus

Question

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Horizontal and passing through the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\)

Step-by-Step Solution

Answer

The equation is \(y = \frac{3}{4}\).

Step 1: Identify the type of line

Since the line is described as horizontal, we know that its slope, denoted as \(m\), is zero. A horizontal line has the form \(y = c\), where \(c\) is a constant.

Step 2: Determine the constant \(c\)

Given the line passes through the point \(\left(\frac{1}{2}, \frac{3}{4}\right)\), we use the y-coordinate of this point as the constant \(c\) in the equation of the line. Therefore, the equation is \(y = \frac{3}{4}\).

Step 3: Write the equation in slope-intercept form

To express the equation in the form \(y = mx + b\), recognize that for a horizontal line, \(m = 0\). Thus, the equation rewrites as \(y = 0 \cdot x + \frac{3}{4}\), simplifying back to \(y = \frac{3}{4}\).

Key Concepts

horizontal line

A horizontal line is a straight line that runs from left to right in a coordinate plane and is parallel to the x-axis. This type of line has a very special characteristic: it never rises or falls as you move along it. Therefore, it has zero vertical change, which results in a slope of zero. When we describe a line as horizontal, we are essentially saying that every point on the line shares the same y-coordinate.
For instance, if a horizontal line passes through a point with a y-coordinate of \( \frac{3}{4} \), every other point on the line also has this same y-coordinate. This leads to the equation of a horizontal line being written simply as \( y = c \), where \( c \) is the constant value of the y-coordinate for all points on that line.

• Horizontal lines have a slope of zero.
• The equation form for horizontal lines is \( y = c \).
• Every point on the line has the same y-coordinate.

slope-intercept form

The slope-intercept form is a way of writing the equation of a line so that you can see both the slope and the y-intercept easily. It is expressed as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept. The y-intercept is the point where the line crosses the y-axis. This form is very convenient for quickly graphing a line and understanding its steepness as well as how high or low it starts.
Despite the simplicity of writing horizontal lines as \( y = c \), converting them into slope-intercept form is still straightforward. However, since the slope \( m \) for a horizontal line is zero, any term involving \( x \) will vanish. Thus, a horizontal line like \( y = \frac{3}{4} \) in slope-intercept form becomes \( y = 0 \cdot x + \frac{3}{4} \), further simplifying back to \( y = \frac{3}{4} \).

• Slope-intercept form is \( y = mx + b \).
• \( m \) represents the slope, showing the tilt or incline of the line.
• \( b \) is the y-intercept where the line meets the y-axis.
• For horizontal lines, \( m = 0 \), so it simplifies to \( y = b \).

slope

The slope of a line is a numerical value that defines its steepness and direction. It can be thought of as the "rise over run," or the amount the line goes up or down as you move from one point to another along its length. The slope is calculated as the change in y-coordinates divided by the change in x-coordinates between two distinct points on the line: \( m = \frac{\Delta y}{\Delta x} \).

• If the result is positive, the line slopes upward.
• If it is negative, the line slopes downward.

For a horizontal line, like the one passing through \( \left( \frac{1}{2}, \frac{3}{4} \right) \), there is no rise, meaning the slope \( m \) is 0. This zero slope guarantees a flat line that does not tilt in either direction.Additionally, since the change in the y-coordinate is zero regardless of the x-coordinate, horizontal lines ensure a uniform y-value.