Problem 36 - 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 \((-1,5)\) and parallel to the horizontal line passing through \((2,-1)\)

Step-by-Step Solution

Answer

The equation of the line is \(0x + y = 5\).

Step 1: Identify Characteristics of the Given Line

The horizontal line passing through the point \((2, -1)\) has a constant \(y\)-value of \(-1\). Since it is horizontal, its slope is \(m = 0\). Parallel lines have the same slope. Thus, the line we are finding also has a slope of \(m = 0\).

Step 2: Determine the Equation in Point-Slope Form

Use the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point \((-1, 5)\) and \(m = 0\). Plugging in these values, we have: \[y - 5 = 0(x + 1)\]. This simplifies to \(y = 5\).

Step 3: Convert the Equation to Standard Form

The standard form of a line is \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) is non-negative. The equation \(y = 5\) can be rewritten in standard form as \(0x + y = 5\).

Key Concepts

Parallel Lines

Parallel lines are a fundamental concept in geometry. They are lines that never meet, regardless of how far they might be extended on a plane. This happens because parallel lines have the same slope, which means they rise and run at the same rate. In simpler terms, if you were to walk along parallel lines, you would be traveling in the same direction and at the same incline.
For a line to be parallel to another, their slopes ( m ) must be equal. For instance, if the slope of one line is 0 because it's horizontal, then a line parallel to it will also have a slope of 0.

• The distance between parallel lines remains constant.
• When graphed, parallel lines appear as equally spaced lines that do not intersect.

This concept is vital for determining the equation of new lines that need to be parallel to existing ones, just like in our original exercise.

Horizontal Line

A horizontal line on a graph is a line that extends from left to right and has a constant y-value for every x-point. This means every point along the line is at the same vertical height, hence the 'horizontal' aspect. Horizontal lines have a slope of 0.
For example, the horizontal line passing through the point (2, -1) has a slope of m = 0 because it doesn't rise or fall; it simply runs parallel to the x-axis.

• These lines are represented in the equation form y = c , where c is the constant y-value.
• A practical feature of horizontal lines is that they make it easy to see that, despite varying x values, the y-value remains unchanged.

This understanding is imperative when tackling problems needing a parallel line that shares the same slope, as seen in our exercise.

Standard Form

The standard form of a line is an organized way to present a linear equation. It's generally represented as Ax + By = C , where:

• A , B , and C are integers.
• The coefficient A is non-negative.
• Usually, A and B are not both zero.

It is often used because it clearly shows the relationship between the x and y variables, making it convenient for identifying intercepts and comparing different lines.
In the given exercise, we started with a point-slope form and converted it to a standard form. This involves rearranging and simplifying your equation to fit Ax + By = C . For instance, a horizontal line like y = 5 can be quickly rewritten as 0x + y = 5 in standard form. Such clarity and structure simplify problem-solving and graphing significantly.