Problem 20 - Calculus for Biology and Medicine

Question

Determine the equation of the line that satisfies the stated requirements. Put the equation in standard form. The line with slope \(-1\) and \(y\) -intercept \((0,-3)\)

Step-by-Step Solution

Answer

The equation in standard form is \(x + y = -3\).

Step 1: Identify Given Values

We know the slope of the line is \(-1\) and the y-intercept is \((0, -3)\). A line with these characteristics can be represented initially in slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Apply Slope-Intercept Form

Substitute the given slope \(-1\) and y-intercept \(-3\) into the slope-intercept form, resulting in: \(y = -x - 3\). This equation is currently in slope-intercept form.

Step 3: Convert to Standard Form

Standard form of a line is written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. To convert \(y = -x - 3\) to standard form, add \(x\) to both sides, obtaining \(x + y = -3\).

Step 4: Verify Standard Form

Ensure the coefficients \(A\), \(B\), and \(C\) are integers and \(A\) is non-negative. The equation \(x + y = -3\) satisfies these conditions, so it is correctly in standard form.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express the equation of a line. It looks like this: \(y = mx + b\). Here:

• \(m\) represents the slope of the line.
• \(b\) stands for the y-intercept, which is where the line crosses the y-axis.

This form is particularly useful for quickly identifying the slope and y-intercept of a line. You can easily determine if a line goes up or down by looking at the sign of the slope:

• A positive slope means the line rises as it moves to the right.
• A negative slope indicates the line falls as it goes to the right.

In our case, the slope is \(-1\), so the line will slant downwards. The y-intercept is the point \((0, -3)\), meaning the line will cross the y-axis three units below the origin.

Standard Form of a Line

The standard form of a line differs from the slope-intercept form by how it structures the equation. It is written as \(Ax + By = C\). In this format:

• \(A\), \(B\), and \(C\) are integers.
• \(A\) should ideally be a non-negative number.

This form is preferable when you want to avoid fractions or when information about both x-intercept and y-intercept might be necessary. To convert from slope-intercept form to standard form, rearrange the equation so that both \(x\) and \(y\) are on the left side, and constants are on the right side. Thus, if you start with \(y = -x - 3\), add \(x\) to both sides to obtain \(x + y = -3\). This places the equation neatly in the standard format, making it easy to analyze different aspects of the line.

Slope of a Line

The slope of a line, symbolized as \(m\) in the slope-intercept form, quantifies the line's steepness or direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line:\[m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}\]The slope is crucial because it tells us several things about the line:

• If \(m > 0\), the line is inclined upwards as it moves from left to right.
• If \(m < 0\), the line slopes downwards.
• If \(m = 0\), the line is horizontal, meaning there is no vertical change as you move along the line.

For the problem given, the slope is \(-1\). This means for every unit the line goes right, it goes down one unit. Hence, the line steadily declines, reflecting its negative slope.