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 standard form of the line is \(0x + y = 5\).

Step 1: Understand the Line Properties

The line is parallel to a horizontal line, indicating it has the same orientation. A horizontal line has a slope of 0, meaning our line should also have a slope of 0 and therefore be horizontal.

Step 2: Recognize the Line Equation Format

Any horizontal line has the form \(y = b\), where \(b\) is the y-coordinate which remains constant for all x-values.

Step 3: Use the Given Point

Our line passes through the point \((1, 5)\). Since it is horizontal, universally, the y-coordinate is always 5. Thus, the equation of the line is \(y = 5\).

Step 4: Convert to Standard Form

The standard form of a line is \(Ax + By = C\). For a horizontal line \(y = 5\), rewrite it in the form \(0x + 1y = 5\). Therefore, the standard form of the line is \(0x + y = 5\).

Key Concepts

Horizontal Lines

Understanding horizontal lines can greatly simplify our comprehension of particular mathematical problems. A horizontal line is uniquely characterized by its zero slope. This means it does not rise or fall as it moves along the x-axis. Every point on this line has the same y-coordinate, reflected in the equation format as \(y = b\). Here, \(b\) represents a constant value, the y-coordinate of the line, signifying that the line remains perfectly flat.

When encountering exercises involving horizontal lines, it's crucial to realize that only the y-value is crucial for defining the line's equation. Whether the line passes through (2, 1) or (1, 5), as cited in our exercise, the horizontal nature of the line dictates that the only determinant is the consistent y-value, like \(y = 5\) in the case of this exercise.

This simple format distinguishes horizontal lines from other line types, offering a straightforward method of ascertaining a line's equation, whatever the x-values encountered may be.

Parallel Lines

Parallel lines play an integral role in various geometry and algebra problems. These lines share identical slopes, meaning they maintain a consistent distance from each other and never intersect. Recognizing parallelism often involves observing the line's slopes; if equal, the lines are parallel.

In the given exercise, we had a scenario where our line needed to be parallel to a horizontal line. Since horizontal lines have a slope of zero, any line parallel to it will also possess a slope of zero. Thus, not only does the equation take the form \(y = b\), but we can also use any different y-coordinates for lines parallel to a given horizontal line, as long as their slope remains zero.

While applying these principles to the exercise, the line passing through (1, 5) is parallel to the line passing through (2, 1), maintaining a slope of zero and indicating the parallel nature with the same flat horizontal orientation.

Standard Form of a Line

In mathematics, expressing equations in various forms enhances our comprehension and manipulation of lines. The standard form of a line is particularly notable, defined by \(Ax + By = C\). Here, \(A\), \(B\), and \(C\) are integers, and this representation elegantly accommodates both vertical, horizontal, and oblique lines.

In the exercise we solved, transforming the slope-intercept form of a horizontal line \(y = 5\) into standard form becomes a straightforward process. Since a horizontal line implies no change in the x-direction, its equation in standard form becomes \(0x + 1y = 5\). Here, the zero multiplier of \(x\) emphasizes the lack of horizontal shift in the line.

Using standard form enhances versatility in solving equations and graphing, providing a consistent methodology for dealing with lines across various mathematical contexts. Ultimately, mastering these transformations augments one's algebraic toolset, enabling clearer problem-solving strategies.