Problem 28 - 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 \((1,2)\) and parallel to \(x-2 y+4=0\)

Step-by-Step Solution

Answer

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

Step 1: Identify slope of original line

The given line is in the form \(x - 2y + 4 = 0\). We want to convert it into slope-intercept form \(y = mx + b\) to find its slope. First, solve for \(y\):\[x - 2y + 4 = 0\]Rearrange to get:\[x + 4 = 2y\]\[y = \frac{1}{2}x + 2\]Hence, the slope \(m\) of the line is \(\frac{1}{2}\).

Step 2: Use point-slope form for the new line

Since the lines are parallel, the new line will also have a slope of \(\frac{1}{2}\). We use the point-slope form \(y - y_1 = m(x - x_1)\) with point \((1, 2)\) and slope \(\frac{1}{2}\):\[y - 2 = \frac{1}{2}(x - 1)\]

Step 3: Simplify the equation

Distribute the slope on the right side:\[y - 2 = \frac{1}{2}x - \frac{1}{2}\]Add 2 to both sides to isolate \(y\):\[y = \frac{1}{2}x + \frac{3}{2}\]

Step 4: Convert to standard form

The standard form of a line equation is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers. Convert the equation \(y = \frac{1}{2}x + \frac{3}{2}\) to this form:First, multiply the entire equation by 2 to eliminate fractions:\[2y = x + 3\]Rearrange to get the equation in standard form:\[x - 2y = -3\]

Key Concepts

Equation of a Line

An equation of a line is a mathematical representation of all the points lying on that line. It helps us understand the relationship between the x and y coordinates on a 2D plane. The general formula to represent a line is \(Ax + By = C\), but it can be expressed in various forms, like slope-intercept or point-slope form, based on different needs.

Each form has its own use cases, making it versatile and handy in different mathematical and real-world applications.

• The slope-intercept form helps quickly identify slope and y-intercept.
• Standard form is ideal for showcasing integer relationships.
• Point-slope form is perfect for constructing a line when you know a slope and a point on the line.

Understanding these forms enables you to solve geometric problems with ease, like identifying parallel lines or computing intersections.

Slope-Intercept Form

Slope-intercept form is one of the most commonly used equations to express a line. It is given by:\(y = mx + b\), where

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

This form is particularly useful because it provides a quick way to graph a line, by allowing us to immediately understand how steep the line is (due to the slope \(m\)) and where it cuts the y-axis.

For example, in our original solution, we converted the line equation \(x - 2y + 4 = 0\) to\(y = \frac{1}{2}x + 2\), revealing a slope of \(\frac{1}{2}\) and a y-intercept of 2. This makes it clear that as \(x\) increases by 1, \(y\) increases by \(\frac{1}{2}\).

Such understanding helps in graphing the line accurately on the coordinate plane, making the slope-intercept form a favorite among students and professionals.

Standard Form

The standard form of a line's equation is expressed as \(Ax + By = C\), where

• \(A\), \(B\), and \(C\) are integers.
• \(A\) and \(B\) are not both zero.

This form is especially useful when working with integer solutions, as it can help simplify finding intersection points of lines or solving systems of linear equations.

For the line in the exercise \(x - 2y = -3\), we can quickly see its coefficients are integers, making it convenient for solving with other similar equations.

Standard form is also preferred for certain types of algebraic operations, helping algebraists find solutions without committing a lot to memory. Connecting standard form and slope-intercept form allows deeper insights into linear relationships and their graphical interpretations.

Point-Slope Form

Point-slope form becomes particularly useful when you know a point on the line and the slope. It's given by:\(y - y_1 = m(x - x_1)\), where

• \((x_1, y_1)\) is a known point.
• \(m\) is the slope of the line.

This form allows you to derive an equation widely from real-world data or from given geometric conditions. In the exercise, we use point-slope form to find an equation of a line parallel to \(x - 2y + 4 = 0\), passing through a specific point (1,2).

We took the known slope \(\frac{1}{2}\) and the known point, and created the line equation \(y - 2 = \frac{1}{2}(x - 1)\).

This form of equation is easy to transform into slope-intercept or standard form, depending on what's needed, showing its flexibility and appeal in mathematical problem-solving.