Problem 10 - 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 passing through \((-3,5)\) with slope \(1 / 2\)

Step-by-Step Solution

Answer

The equation of the line in standard form is \( x - 2y = -13 \).

Step 1: Use the Point-Slope Formula

The point-slope form of a line equation is given by \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Substitute \((-3, 5)\) for \((x_1, y_1)\) and \(\frac{1}{2}\) for \(m\) to get: \( y - 5 = \frac{1}{2}(x + 3) \).

Step 2: Simplify the Equation

Distribute \(\frac{1}{2}\) on the right side of the equation: \( y - 5 = \frac{1}{2}x + \frac{1}{2} \times 3 \). This simplifies to \( y - 5 = \frac{1}{2}x + \frac{3}{2} \).

Step 3: Isolate the Variable Terms

Add 5 to both sides to move constants to one side: \( y = \frac{1}{2}x + \frac{3}{2} + 5 \). Convert 5 to \(\frac{10}{2}\) for easy addition: \( y = \frac{1}{2}x + \frac{3}{2} + \frac{10}{2} \).

Step 4: Combine Terms

Now combine the fractions: \( y = \frac{1}{2}x + \frac{13}{2} \).

Step 5: Convert to Standard Form

Multiply every term by 2 to eliminate fractions: \( 2y = x + 13 \). Rearrange to standard form \( Ax + By = C \): subtract \(x\) from both sides to get \( -x + 2y = 13 \). Multiply by -1 to have the x-term positive, resulting in \( x - 2y = -13 \).

Key Concepts

Linear Equations

A linear equation is a fundamental concept in algebra. It represents a straight line when graphed on a coordinate plane. The general form of a linear equation can be expressed as \( y = mx + b \), where:

• \(m\) is the slope that shows the steepness or the inclination of the line.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

A linear equation can also have other forms, such as point-slope form and standard form. Regardless of the form, all linear equations create lines relative to the graph's axes. These equations are incredibly useful for understanding relationships between variables, forecasting trends, and solving various real-world problems.
In the exercise at hand, we start with the point-slope form, which allows us to use a given point and the line's slope to construct the equation of the line.

Standard Form of a Line

The standard form of a line is written as \( Ax + By = C \), where:

• \(A\), \(B\), and \(C\) are integers (whole numbers).
• Typically, \(A\) is a positive integer.

The standard form is particularly beneficial for analyzing and comparing different linear equations, as it presents a clear way to see the coefficients of both variables involved.
In solving our exercise, the final step involved transforming the equation from point-slope to standard form. The transformation involves ensuring that all coefficients are integers and usually making \(A\) positive for clarity and convention. So, our equation is transformed into \(x - 2y = -13\), which matches the standard form structure.

Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as \( y = mx + b \). This is often the most straightforward form for understanding how changes in \(x\) affect \(y\):

• \(m\) here indicates the line's slope.
• \(b\) indicates where the line crosses the y-axis.

This form is exceptionally useful for quickly identifying the slope and intercept, which are crucial for graphing and interpreting linear relationships. In the exercise, once we derived the equation using the point-slope form, it was simplified to \( y = \frac{1}{2}x + \frac{13}{2} \) which is in slope-intercept form.
This version makes it easy to plot the line or predict values of \(y\) for given \(x\). However, for the completeness of the exercise, the equation was eventually converted into standard form.