Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, find an equation for each line. Then write your answer in the form \(A x+B y+C=0 .\) Through \((2,3)\) and \((4,8)\)a

Step-by-Step Solution

Answer

The equation in standard form is \(5x - 2y - 4 = 0\).

Step 1: Calculate the Slope

First, find the slope \( m \) of the line using the two given points \((2, 3)\) and \((4, 8)\). The formula for the slope \( m \) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \( (x_1, y_1) = (2, 3) \) and \( (x_2, y_2) = (4, 8) \). Thus, \( m = \frac{8 - 3}{4 - 2} = \frac{5}{2} \).

Step 2: Use Point-Slope Form

Use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \), to write the equation. We will use the point \((2, 3)\). Substituting, we get \[ y - 3 = \frac{5}{2}(x - 2) \].

Step 3: Simplify and Write in Slope-Intercept Form

Expand and simplify the equation from Step 2: \[ y - 3 = \frac{5}{2}x - 5 \]. Adding 3 to both sides results in \[ y = \frac{5}{2}x - 2 \]. This is the slope-intercept form.

Step 4: Convert to Standard Form

To convert \( y = \frac{5}{2}x - 2 \) to standard form \( Ax + By + C = 0 \), start by eliminating the fraction by multiplying the entire equation by 2 to get \( 2y = 5x - 4 \). Rearrange to obtain \( 5x - 2y - 4 = 0 \), which is in the desired form.

Key Concepts

Slope Calculation

Finding the slope is the first essential step in writing the equation of a line. Slope, often represented as \( m \), tells us how steep the line is. To calculate the slope, you need at least two points on the line. The slope is defined by the formula:

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

This formula measures the change in \( y \) divided by the change in \( x \) or, in simpler terms, "rise over run." From our example, using the points \((2, 3)\) and \((4, 8)\), the slope is determined as follows:

• \( m = \frac{8 - 3}{4 - 2} = \frac{5}{2} \)

This result shows that for every 2 units you move along the x-axis, you will move 5 units along the y-axis. A positive slope indicates a line rising from left to right.

Point-Slope Form

The point-slope form is a reliable method to create an equation when you know a point on the line and its slope. The formula for the point-slope form is:

• \( y - y_1 = m(x - x_1) \)

Using this form allows you to plug in the slope and the coordinates of the known point seamlessly. Let’s see this in action using our example, where the slope \( m = \frac{5}{2} \) and the known point is \((2, 3)\):

• \( y - 3 = \frac{5}{2}(x - 2) \)

This format directly shows how the line inclines or declines from a specific point. It is particularly useful for understanding how the slope affects the line's angle and direction.

Slope-Intercept Form

Once you have the point-slope form, converting it to the slope-intercept form, \( y = mx + b \), gives you another popular equation format. This form makes it easy to see the slope (\( m \)) and the y-intercept (\( b \)), where the line crosses the y-axis. To simplify from our point-slope form:

• \( y - 3 = \frac{5}{2}x - 5 \)

Add 3 to both sides to get:

• \( y = \frac{5}{2}x - 2 \)

Here, you can easily identify the slope \( \frac{5}{2} \) and see that the line crosses the y-axis at -2. This form is excellent for quickly sketching graphs since it provides a clear starting and gradual incline.

Standard Form

The standard form of a line's equation is \( Ax + By + C = 0 \), which is another format that's sometimes more convenient for solving systems of equations. To convert from slope-intercept to standard form, you need to rearrange and clear any fractions if necessary. Starting with:

• \( y = \frac{5}{2}x - 2 \)

Multiply every term by 2 to eliminate the fraction:

• \( 2y = 5x - 4 \)

Reordering terms gives:

• \( 5x - 2y - 4 = 0 \)

This format highlights the coefficients of both \( x \) and \( y \), which can be helpful for various algebraic manipulations, such as finding intersections with other lines.