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 \((2,-1)\) and parallel to the line passing through \((0,-4)\) and \((2,1)\)

Step-by-Step Solution

Answer

The equation of the line is \( 5x - 2y = 12 \).

Step 1: Determine the slope of the given line

To find the slope of the line passing through the points \((0,-4)\) and \((2,1)\), use the slope formula: \( m = \frac{y_2-y_1}{x_2-x_1} \). Here, \((x_1, y_1) = (0, -4)\) and \((x_2, y_2) = (2, 1)\), so \( m = \frac{1 - (-4)}{2 - 0} = \frac{5}{2} \). Thus, the slope of this line is \( \frac{5}{2} \).

Step 2: Use the slope to find the parallel line's equation

Since the line we need is parallel to the given line, it will have the same slope, \( \frac{5}{2} \). Using the point-slope form of the equation \( y - y_1 = m(x - x_1) \) with the point \((2, -1)\): \( y + 1 = \frac{5}{2}(x - 2) \).

Step 3: Simplify to slope-intercept form if needed

Distribute the slope across the terms: \( y + 1 = \frac{5}{2}x - 5 \). Then subtract 1 from both sides to isolate \( y \): \( y = \frac{5}{2}x - 6 \).

Step 4: Convert to standard form

To convert the slope-intercept form \( y = \frac{5}{2}x - 6 \) to standard form, eliminate the fraction by multiplying the entire equation by 2: \( 2y = 5x - 12 \). Rearrange into \( Ax + By = C \) form by moving terms around: \( 5x - 2y = 12 \), which is the standard form.

Key Concepts

Slope Formula

The slope formula is a fundamental concept when dealing with straight lines. It helps us calculate how steep a line is or how it inclines. To determine the slope between two points, we use the formula:

• \( m = \frac{y_2-y_1}{x_2-x_1} \)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. The difference in the \(y\)-coordinates is divided by the difference in the \(x\)-coordinates.
The result is the slope \(m\), which tells us how much the line rises for every unit it runs horizontally.
If the slope is positive, the line rises as it moves from left to right, while a negative slope means it falls.

Point-Slope Form

The point-slope form of a line's equation is very useful when we know a point on the line and the slope. If a line passes through a point \((x_1, y_1)\) and has a slope \(m\), its equation can be written as:

• \( y - y_1 = m(x - x_1) \)

This form allows us to quickly build the equation of a line, as it incorporates both the slope and an existing point.
To find the line parallel to another, we maintain the slope and use a different point.
For example, using the slope \(\frac{5}{2}\) and point \((2, -1)\), we get:

• \( y + 1 = \frac{5}{2}(x - 2) \)

This form gives us a starting point to later convert into other forms if required.

Parallel Lines

Parallel lines are lines in a plane separated by the same distance and never touch. They have equal slopes.
If given a line and a point, to find a parallel line through that point, simply use the same slope with the new point in the point-slope form.
Parallel lines feature prominently in mathematics due to their consistency and predictability, especially in coordinate geometry.
For instance, if a line goes through points \((0, -4)\) and \((2, 1)\), with slope \(\frac{5}{2}\), any line parallel to it, such as the one through \((2, -1)\), will share this slope, reinforcing their unending parallel nature.

Standard Form

The standard form of a line's equation makes it easier to analyze and compare different lines. It's typically expressed as:

• \( Ax + By = C \)

where \(A\), \(B\), and \(C\) are integers and \(A\) should be positive.
To convert any line equation into standard form, you may need to rearrange and eliminate fractions.
Starting from the slope-intercept form \(y = \frac{5}{2}x - 6\), we multiply through by 2 to get:

• \(2y = 5x - 12\)

Then rearrange to standard form:

• \(5x - 2y = 12\)

This form is especially beneficial as it directly shows the coefficients essential for some mathematical computations like intersections.