Problem 19 - 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 3 and \(y\) -intercept \((0,2)\)

Step-by-Step Solution

Answer

The equation is \( 3x - y = -2 \) in standard form.

Step 1: Understand the Standard Equation of a Line

The standard form of a linear equation is given by \( Ax + By = C \). Our goal is to rewrite the line's equation, initially in slope-intercept form, into this form.

Step 2: Start with the Slope-Intercept Form

A line with a slope of 3 and a \( y \)-intercept at point \((0, 2)\) can be written in slope-intercept form as \( y = mx + b \). Here, \( m = 3 \) and \( b = 2 \), thus the equation is \( y = 3x + 2 \).

Step 3: Rearrange to Standard Form

To express \( y = 3x + 2 \) in standard form \( Ax + By = C \), we need \( A \), \( B \), and \( C \) to be integers where \( A \) is positive. First, subtract \( 3x \) from both sides: \( -3x + y = 2 \). To make \( A \) positive, multiply the whole equation by \(-1\): \( 3x - y = -2 \).

Step 4: Verify the Standard Form

Check that the equation \( 3x - y = -2 \) fulfills the requirements. It is in standard form because it corresponds to \( Ax + By = C \) with \( A = 3 \), \( B = -1 \), and \( C = -2 \).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation provides a straightforward way to express straight lines. This form is written as \( y = mx + b \), where:

• \( m \) is the slope of the line, representing the change in \( y \) for each unit change in \( x \).
• \( b \) is the \( y \)-intercept, which is the point where the line crosses the \( y \)-axis.

This form is particularly useful because it gives immediate insight into the behavior of the line. You can easily identify how steep the line is from \( m \) and where it starts on the \( y \)-axis from \( b \). For instance, in the given exercise, the line with a slope of 3 and a \( y \)-intercept of 2 is written as \( y = 3x + 2 \). From this equation, you can see that every time \( x \) increases by 1, \( y \) will increase by 3. Thus, the slope-intercept form not only sets up the equation but also offers a visual grasp of the line's nature.

Linear Equation

A linear equation is an algebraic equation involving linear functions. Essentially, it describes a straight line when graphed on a coordinate plane. These equations can be expressed in various forms, including standard form \( Ax + By = C \) and slope-intercept form \( y = mx + b \). The general characteristic of a linear equation is that it maintains a constant rate of change, which translates to a straight line on a graph. There are no curves, parabolas, or differing slopes because each term is either raised to the power of 1 or contains no variable factors at all. This equation type is integral in many mathematical and real-world applications, such as calculating distances, predicting outcomes, and solving for unknown variables when relationships are linear. Understanding how to transition equations between forms and interpret their graph representations is crucial for mastering mathematics.

Slope and Y-Intercept

The slope and \( y \)-intercept are key components of a line's equation in the slope-intercept form. The slope, denoted as \( m \), indicates the steepness and direction of the line. A positive slope means the line ascends from left to right, while a negative slope descends.

• A larger absolute value of the slope means a steeper line.
• A slope of 0 results in a horizontal line, which is parallel to the \( x \)-axis.

The \( y \)-intercept, represented as \( b \), indicates where the line meets the \( y \)-axis. In our problem, the line has a slope of 3 and a \( y \)-intercept of 2, implying it crosses the \( y \)-axis at the point \((0, 2)\). This point is critical because it provides a starting point for graphing the line. In summary, both the slope and \( y \)-intercept are fundamental in constructing the full picture of a linear equation's graph.