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 \(3x - y = -2\).

Step 1: Identify the Slope-Intercept Form

The slope-intercept form of a line's equation is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = 3 \) and \( b = 2 \).

Step 2: Substitute the Values

Plug the given slope and y-intercept into the slope-intercept form equation. This gives us \( y = 3x + 2 \).

Step 3: Rearrange to Standard Form

The standard form of a line's equation is \( Ax + By = C \). To convert, subtract \( y \) from both sides of \( y = 3x + 2 \) to get \( 0 = 3x - y + 2 \).

Step 4: Simplify the Equation

Rearrange the equation to align with \( Ax + By = C \), resulting in \( 3x - y = -2 \). Now the equation is in standard form.

Key Concepts

Slope-Intercept Form

The Slope-Intercept Form is a way to express the equation of a line simply and directly. It is written as \( y = mx + b \). Here, \( y \) represents the dependent variable or output, \( m \) is the slope of the line, \( x \) is the independent variable or input, and \( b \) is the y-intercept. The y-intercept is where the line crosses the y-axis.

• The slope \( m \) describes how steep the line is and in which direction it tilts. If \( m \) is positive, the line rises as it moves left to right. If negative, it falls.
• The y-intercept \( b \) is the starting value of \( y \) when \( x \) is zero. It's the point \((0, b)\) on the graph.

Knowing \( m \) and \( b \), you can immediately sketch out the line. This form makes it easy to identify the characteristics of the line quickly.

Standard Form

Standard Form is another way to express the equation of a line. It is framed as \( Ax + By = C \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative number. Here’s what you should know:

• This form is a bit more flexible when dealing with nondiscrete intersections or varied line relations.
• Standard form is often preferred for solving systems of linear equations and finding intersections since you can manipulate equations more easily.
• It requires a specific setup: typically, \( A \), \( B \), and \( C \) are kept to their simplest integer ratio.

Transforming from slope-intercept to standard form may involve rearranging and simplifying the terms, ultimately illustrating the relationship between \( x \) and \( y \) differently.

Equation of a Line

The equation of a line connects linear algebra to geometry by describing a straight line in a 2-dimensional space. Using the slope-intercept or standard form, you essentially convey the same linear relationship; they just offer different insights.

• In the slope-intercept form, emphasis is on the slope and starting point \( (0, b) \). It’s great for straightforward graphing and immediate analysis of how the line behaves.
• By contrast, the standard form handles algebraic manipulation better, especially with systems of equations.
• Both formats can be transformed into one another wherein the slope \( m \) is computed as \( -\frac{A}{B} \) from standard form, and vice versa for conversions.

Equations of a line offer a versatile tool set for representing, analyzing, and solving real-world problems using theoretical and graphical insights.