Problem 34 - 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 \((4,-1)\) and perpendicular to the line passing through \((-2,0)\) and \((1,1)\)

Step-by-Step Solution

Answer

The equation of the line is \(3x + y = 11\).

Step 1: Find the Slope of the Given Line

First, calculate the slope of the line passing through the points \((-2,0)\) and \((1,1)\). Use the slope formula: \(m = \frac{y_2-y_1}{x_2-x_1}\). Substituting in the given points, the slope \(m\) is \(\frac{1-0}{1-(-2)} = \frac{1}{3}\).

Step 2: Determine the Slope of the Perpendicular Line

Lines that are perpendicular to each other have slopes that are negative reciprocals. So, if the slope of the original line is \(\frac{1}{3}\), the slope of the perpendicular line is \(-3\).

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

Using the point-slope form of the equation \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is the point \((4,-1)\), we substitute the values to get: \(y + 1 = -3(x - 4)\).

Step 4: Simplify to Standard Form

Expand and simplify the equation: \(y + 1 = -3x + 12\). Rearrange this to the standard form \(Ax + By = C\), giving \(3x + y = 11\).

Key Concepts

Understanding the Slope of a Line

The slope of a line is a fundamental concept in geometry and algebra that measures how steep a line is. Imagine you are sitting at the top of a hill. The slope tells you how steep this hill is. Mathematically, the slope is represented as "\(m\)" and is calculated using the difference in the \(y\)-coordinates divided by the difference in the \(x\)-coordinates between two points on a line. The formula is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

Here, \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of any two points on the line.
Using this formula, if we substitute the points \((-2, 0)\) and \((1, 1)\), the slope \(m\) turns out to be \(\frac{1}{3}\). This means for every three units right across in horizontal direction, the line moves one unit up in the vertical direction.
This slope not only tells us how the line climbs or falls but also plays a crucial role when it comes to finding perpendicular lines. Perpendicular lines have negative reciprocal slopes. Thus, if a line has a slope of \(\frac{1}{3}\), the perpendicular line will have a slope of \(-3\).

Explaining the Point-Slope Form

The point-slope form of a line's equation is a powerful tool when you know a point on the line and the slope. This form is expressed as:

• \(y - y_1 = m(x - x_1)\)

Here, \(m\) is the slope of the line, and \((x_1, y_1)\) is a specific point on the line.
It's a versatile form because if you know the slope and one point through which the line passes, you can fully describe the line with this equation. For instance, if you've determined that the slope of the line you are interested in is \(-3\) and it passes through the point \((4, -1)\), you can plug these into the formula to get:

• \(y + 1 = -3(x - 4)\)

This equation describes a line slanting downwards because of its negative slope and running through the specified point.

Simplifying to the Standard Form of a Line

Once you have your equation in the point-slope form, you can rearrange it to what's known as the standard form for a line. Standard form looks like this:

• \(Ax + By = C\)

Where \(A\), \(B\), and \(C\) are integers, and \(A\) should be non-negative.
To convert from point-slope form, like \(y + 1 = -3(x - 4)\), first distribute and simplify:

• \(y + 1 = -3x + 12\)

Then, arrange terms to shift \(x\) and \(y\) to one side:

• \(3x + y = 11\)

The result is a neat, concise representation of the line: \(3x + y = 11\) in standard form, which clearly shows one of the infinite set of lines satisfying the given conditions.