Problem 30 - 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 \((2,-3)\) and parallel to the line passing through \((0,-1)\) and \((2,1)\)

Step-by-Step Solution

Answer

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

Step 1: Find the Slope of the Given Line

To find the slope of the line passing through the points \( (0,-1) \) and \( (2,1) \), use the formula for slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) = (0, -1) \) and \( (x_2, y_2) = (2, 1) \). Plug these values into the formula: \[ m = \frac{1 - (-1)}{2 - 0} = \frac{2}{2} = 1 \] Thus, the slope of the line is \ m = 1 \.

Step 2: Identify the Slope of the Parallel Line

Since parallel lines have identical slopes, the slope of the line parallel to the one passing through \( (0, -1) \) and \( (2, 1) \) is also \( m = 1 \). Thus, the slope of the line we need to find is \( 1 \).

Step 3: Use Point-Slope Form to Find the Equation

Use the point-slope form of a line equation \( y - y_1 = m(x - x_1) \), where \( m = 1 \) and the point given is \( (2, -3) \). Substitute these into the equation: \[ y - (-3) = 1(x - 2) \]y + 3 = x - 2. \

Step 4: Rearrange to Standard Form

To convert the equation \( y + 3 = x - 2 \) to standard form \( Ax + By = C \), you need to move all terms to one side: \[ y + 3 = x - 2 \ y - x = -2 - 3 \ -x + y = -5 \] Multiply the entire equation by \( -1 \) to adjust the coefficients so that the x-term is positive: \[ x - y = 5 \] This is the standard form.

Key Concepts

Slope

The concept of a "slope" is central when we talk about lines in algebra and geometry. In simple terms, the slope tells us how steep a line is. It is essentially the rate at which a line rises vertically compared to its horizontal movement. To find the slope of a line given two points,

• Identify the coordinates of the points, such as \( (x_1, y_1) \) and \( (x_2, y_2) \).
• Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula calculates the "rise" over the "run" between the two points.The slope can tell us a lot:

• A positive slope indicates that the line goes upwards.
• A negative slope means the line goes downwards.
• A zero slope implies a horizontal line, whereas an undefined slope indicates a vertical line.

Point-Slope Form

The "point-slope form" of a line equation is a very handy way to write an equation when you know the slope of a line and one point on the line. This form is written as \[ y - y_1 = m(x - x_1) \]

• Here, \( m \) is the slope.
• The coordinates \( (x_1, y_1) \) represent the known point on the line.

This formula helps bridge the gap between having just numerical data and generating an actual equation for a line. It's an easy way to derive the equation of a line when you have partial information. It emphasizes how the slope applies to any point along the line thus making it quite flexible to use. Once you plug in values and calculate, you can convert the result into other forms like the standard form.

Parallel Lines

"Parallel lines" are lines that, when drawn in a plane, never intersect each other. This characteristic of non-intersection is because they share the same slope. Therefore, knowing the defining property of parallel lines helps in finding unknown equations quickly. Here’s why slope is so crucial:

• If two lines are parallel, their slopes are the same.
• No matter where these lines are plotted in a coordinate system, if their slopes match, they will never meet.

When tasked with finding a line parallel to a given line, simply determine the slope of the existing line and use it for the new line. This is due to the fact that having the same slope guarantees that the two lines will not cross paths. In this exercise, identifying the slope of the base line allowed us to form the equation of another line to which it is parallel.