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 \((4,8)\)

Step-by-Step Solution

Answer

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

Step 1: Calculate the Slope of the Line

To find the equation of a line, we first need its slope. Use the formula for slope, which is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For the points \((2,3)\) and \((4,8)\), we have \(m = \frac{8 - 3}{4 - 2} = \frac{5}{2}\). Thus, the slope \(m\) is \(\frac{5}{2}\).

Step 2: Use Point-Slope Formula

With the slope \(m = \frac{5}{2}\) and a point \((2,3)\), we can use the point-slope form to write the equation of the line: \(y - y_1 = m(x - x_1)\). Substituting the values, we get: \(y - 3 = \frac{5}{2}(x - 2)\).

Step 3: Simplify the Equation

Distribute \(\frac{5}{2}\) in the equation: \(y - 3 = \frac{5}{2}x - 5\). Simplify this to \(y = \frac{5}{2}x - 2\).

Step 4: Convert to Standard Form

To convert the equation \(y = \frac{5}{2}x - 2\) into the form \(A x + B y + C = 0\), we multiply through by 2 to eliminate the fraction: \(2y = 5x - 4\). Rearrange terms to get the equation in standard form: \(5x - 2y - 4 = 0\).

Key Concepts

Slope Calculation

Calculating the slope is the first essential step in understanding the equation of a line. The slope describes the steepness and direction of the line. It is represented by the letter \(m\) and can be calculated using two points that the line passes through. Consider two points on the line: \((x_1, y_1)\) and \((x_2, y_2)\). The formula for calculating the slope \(m\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]This formula gives you the rise (change in \(y\)) over the run (change in \(x\)). For example, given the points \((2, 3)\) and \((4, 8)\), the change in \(y\) is \(8 - 3 = 5\) and the change in \(x\) is \(4 - 2 = 2\). Therefore, the slope becomes \(\frac{5}{2}\).

• The larger the slope, the steeper the line.
• A positive slope indicates the line goes up from left to right.
• A negative slope indicates the line goes down from left to right.

Point-Slope Form

The point-slope form of a linear equation helps us find the equation of a line given a slope and a point on the line. It is especially useful when you know one point and the slope. The general format of the point-slope form is:\[ y - y_1 = m(x - x_1) \]Where \((x_1, y_1)\) is a point on the line and \(m\) is the slope previously calculated. This equation is straightforward to use. Simply substitute the values of the known point and the slope.For instance, with a slope of \(\frac{5}{2}\) and the point \((2, 3)\), you substitute these into the formula to get:\[ y - 3 = \frac{5}{2}(x - 2) \]The point-slope form provides a quick and clear way to write the equation of a line, and from here, you can convert it into other forms as needed.

Standard Form for Linear Equations

The standard form for a linear equation is another way to represent a line mathematically. It is typically written as \( Ax + By + C = 0 \), where \( A \), \( B \), and \( C \) are integers, and \( A \) should ideally be positive. This format is quite versatile and used widely in algebra.To convert from the point-slope form \(y - 3 = \frac{5}{2}(x - 2)\) to the standard form, follow these steps:

• First, simplify the equation: distribute \(\frac{5}{2}\) to get \(y = \frac{5}{2}x - 2\).
• To eliminate the fraction, multiply the entire equation by 2: \(2y = 5x - 4\).
• Rearrange to achieve the standard form by getting all terms on one side: \(5x - 2y - 4 = 0\).

This arrangement provides an equation where you can easily identify the coefficients \( A = 5 \), \( B = -2 \), and \( C = -4 \). It allows for straightforward comparison of how multiple lines relate to each other across a coordinate plane.