Problem 50 - Calculus with Concepts in Calculus

Question

Find a two-point equation of the given line. The line containing \((3,4)\) and \((1,3)\)

Step-by-Step Solution

Answer

The equation of the line is \(y = \frac{1}{2}x + \frac{5}{2}\).

Step 1: Determine the Slope

To find the equation of the line, we first need to determine the slope (\(m\)). The slope is calculated using the formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Let's use the points \((x_1, y_1) = (3, 4)\) and \((x_2, y_2) = (1, 3)\). So,\[m = \frac{3 - 4}{1 - 3} = \frac{-1}{-2} = \frac{1}{2}\]The slope of the line is \(\frac{1}{2}\).

Step 2: Use the Point-Slope Form

Next, we use the point-slope form of the equation of a line, which is \(y - y_1 = m(x - x_1)\). We can use one of the points we have, let's use \((3,4)\) and the slope \(\frac{1}{2}\) to form the equation.Substitute these values into the point-slope equation:\[y - 4 = \frac{1}{2}(x - 3)\]

Step 3: Convert to Slope-Intercept Form

To express the equation in the slope-intercept form \(y = mx + b\), distribute and simplify the point-slope equation:\[y - 4 = \frac{1}{2}x - \frac{3}{2}\]Add 4 to both sides to solve for \(y\):\[y = \frac{1}{2}x + \frac{5}{2}\]This is the equation in slope-intercept form.

Key Concepts

Slope of a Line

The slope of a line is a measure of how steep the line is, described as the ratio of the vertical change to the horizontal change between two distinct points on the line. It is typically denoted by the letter \( m \). The formula to calculate the slope \( m \) between two points, \((x_1, y_1)\) and \((x_2, y_2)\), is given by:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In simpler terms, the slope tells us how many units we move up or down for every unit we move right or left. For instance, in our example, the points are \((3,4)\) and \((1,3)\). Substituting these into the slope formula gives:

• \( m = \frac{3 - 4}{1 - 3} = \frac{-1}{-2} = \frac{1}{2} \)

This means for every 2 units moved horizontally, the line moves 1 unit vertically upward, hence a positive slope.

Point-Slope Form

The point-slope form of a linear equation is an incredibly useful way to write the equation of a line. It is especially handy when you know a point on the line and its slope. This form is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \((x_1, y_1)\) is a known point, and \( m \) is the slope of the line. Using the point \((3,4)\) from our previous example and the calculated slope \(\frac{1}{2}\), we plug these values into the formula:

• \( y - 4 = \frac{1}{2}(x - 3) \)

This equation directly gives us a line equation expressed relative to a specific point, making it easier to describe or graph the line.

Slope-Intercept Form

The slope-intercept form is a common and straightforward way to express the equation of a line. It has the standard formula:

• \( y = mx + b \)

In this form, \( m \) is the slope of the line, and \( b \) is the y-intercept, where the line crosses the y-axis. To convert from the point-slope form \( y - 4 = \frac{1}{2}(x - 3) \) to slope-intercept form, we need to simplify it:

• First, distribute \( \frac{1}{2} \) to \( (x - 3) \): \( y - 4 = \frac{1}{2}x - \frac{3}{2} \)
• Then, add 4 to both sides to isolate \( y \): \( y = \frac{1}{2}x + \frac{5}{2} \)

Now, the equation is in a clear format, and you can easily identify the slope \( \frac{1}{2} \) and the y-intercept \( \frac{5}{2} \), which helps in sketching the graph of the line quickly.