Problem 19 - Thomas Calculus

Question

In Exercises 17–30, write an equation for each line described. Passes through \((3,4)\) and \((-2,5)\)

Step-by-Step Solution

Answer

Equation: \( y = -\frac{1}{5}x + \frac{23}{5} \)

Step 1: Identify the Formula for the Line

To find the equation of a line given two points, we use the formula for the slope-intercept form of a line, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the Slope (m)

The slope \( m \) can be calculated using the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Substituting the given points \((3,4)\) and \((-2,5)\), we have:\[m = \frac{5 - 4}{-2 - 3} = \frac{1}{-5} = -\frac{1}{5}\]

Step 3: Use the Point-Slope Form of a Line

With the slope \( m = -\frac{1}{5} \) and one of the points, say \((3,4)\), we can find the equation using the point-slope form:\[y - y_1 = m(x - x_1)\]Plugging in the values, we get:\[y - 4 = -\frac{1}{5}(x - 3)\]

Step 4: Convert to Slope-Intercept Form

To convert the point-slope equation to slope-intercept form, simplify and solve for \( y \):\[y - 4 = -\frac{1}{5}x + \frac{3}{5}\]\[y = -\frac{1}{5}x + \frac{3}{5} + 4\]Convert \( 4 \) to a fraction with the same denominator:\[y = -\frac{1}{5}x + \frac{3}{5} + \frac{20}{5}\]\[y = -\frac{1}{5}x + \frac{23}{5}\]

Step 5: Write the Final Equation

The equation of the line in slope-intercept form is:\[y = -\frac{1}{5}x + \frac{23}{5}\]

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward method to express the equation of a line. This form is helpful because it directly displays the slope and y-intercept, making it easy to graph the line or understand its basic properties. The slope-intercept form is written as:

• y = mx + b

Here, m represents the slope of the line, which shows how steep the line is. Meanwhile, b describes the y-intercept, which is the point where the line crosses the y-axis.
Using this form, finding how a line behaves just takes seconds. You first determine the slope and y-intercept from a given equation, and you're ready to plot.
If you're given a point and a slope, you can also convert from the point-slope form to the slope-intercept form to make identifying the y-intercept easier. This conversion involves solving for y and rearranging terms to match the slope-intercept format. Understanding and using this form is essential for mastering linear equations.

Point-Slope Form

When you have a line passing through a specific point and know the slope, the point-slope form is your go-to equation. This form comes in handy when you don't immediately have the y-intercept. Instead, it focuses on a given point on the line defined by its coordinates \(x_1, y_1\) and the slope \(m\).

• The equation is written as: \(y - y_1 = m(x - x_1)\)

This setup means you adjust the equation based on the slope and a single known point.
For example, if a line passes through the point \(3, 4\) with a slope of \(-\frac{1}{5}\), the equation would be \(y - 4 = -\frac{1}{5}(x - 3)\).
From there, you can solve for other variables or convert to slope-intercept form if needed.
The point-slope form is crucial for situations requiring quick graphing of a line given only partial information. It's particularly useful in exercises where you need to write equations quickly or understand the relationship between a specific set of coordinates and the slope.

Slope Calculation

Calculating the slope of a line is an essential skill in understanding linear relationships. The slope indicates the direction and steepness of a line, which tells us how much the line rises or falls as we move along it.

• The formula to find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

The numerator \(y_2 - y_1\) represents the change in the y-values (vertical change), while the denominator \(x_2 - x_1\) stands for the change in the x-values (horizontal change).
For instance, given the points \((3, 4)\) and \((-2, 5)\), the slope would be calculated as \(-\frac{1}{5}\). This negative value illustrates that for every 5 units you move horizontally, the line descends 1 unit vertically.
Understanding slope helps predict how a line behaves and is an integral part of forming equations, regardless if it's the slope-intercept or point-slope form.