Problem 34 - 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 \((4,1)\) and \((8,2)\)

Step-by-Step Solution

Answer

The equation of the line is \(x - 4y = 0.\)

Step 1: Find the Slope

First, we need to find the slope of the line passing through the points \((4,1)\) and \((8,2)\). The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].So, \[ m = \frac{2 - 1}{8 - 4} = \frac{1}{4}. \] Thus, the slope of the line is \(\frac{1}{4}.\)

Step 2: Use Point-Slope Form

Next, we'll use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \(m\) is the slope and \((x_1, y_1)\) is one of the points. Let's use point \((4, 1)\):\[ y - 1 = \frac{1}{4}(x - 4). \]

Step 3: Simplify to Slope-Intercept Form

Now, let's simplify the equation obtained from the point-slope form:\[ y - 1 = \frac{1}{4}x - 1. \]Adding 1 to both sides gives us:\[ y = \frac{1}{4}x. \]

Step 4: Convert to General Form

Convert the slope-intercept form to the general form \(Ax + By + C = 0\). Our current equation is \(y = \frac{1}{4}x\). Let's move all terms to one side:\[ -\frac{1}{4}x + y = 0. \]To eliminate the fraction, multiply the entire equation by 4:\[ -x + 4y = 0. \]

Step 5: Rearrange to Standard Form

The standard form usually has a positive \(A\) value, so rearrange our equation:\[ x - 4y = 0. \]This is the equation of the line in standard form \(Ax + By + C = 0\) with \(A = 1\), \(B = -4\), and \(C = 0\).

Key Concepts

Slope Formula

The slope formula is essential for understanding how steep a straight line is, connecting two points on a graph. It is calculated as the change in the vertical direction (y-coordinates) divided by the change in the horizontal direction (x-coordinates). The formula is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where

• \(x_1, y_1\) are the coordinates of the first point, and
• \(x_2, y_2\) are the coordinates of the second point.

For example, using two points like \((4, 1)\) and \((8, 2)\), we find the slope as follows: \[ m = \frac{2 - 1}{8 - 4} = \frac{1}{4} \] This tells us that for every 4 units we move horizontally, we move 1 unit vertically. Understanding slopes is crucial, as it gives insight into the direction and steepness of a line.

Point-Slope Form

The point-slope form of a linear equation is a handy tool when you know the slope of a line, and at least one point through which the line passes. Its formula is: \[ y - y_1 = m(x - x_1) \] Here,

• \(m\) is the slope, and
• \(x_1, y_1\) is a known point on the line.

Using our earlier example, and the point \((4, 1)\), if the slope is \(\frac{1}{4}\), the equation becomes: \[ y - 1 = \frac{1}{4}(x - 4) \] This form is quite useful for quickly finding a linear equation from a given slope and point. It sets a precise relationship between the slope and any point on the line.

Slope-Intercept Form

Slope-intercept form is among the most popular forms of linear equations. It is structured as: \[ y = mx + b \] where

• \(m\) represents the slope of the line, and
• \(b\) is the y-intercept— the point where the line crosses the y-axis.

After reformulating from the point-slope form, you'll arrive at this form. Using the previous example, we simplified: \[ y - 1 = \frac{1}{4}x - 1 \] Adding 1 gives us: \[ y = \frac{1}{4}x \] This shows that the line doesn't intercept the y-axis at a specific point except at the origin in this simplified example. The slope-intercept format is crucial for easily identifying the slope and intercept directly in an equation.

Standard Form of a Line

The standard form of a linear equation is: \[ Ax + By + C = 0 \] This is particularly useful for analyzing and solving linear systems of equations, as it points out integer values for each coefficient:

• \(A\), \(B\), and \(C\) are integers, with \(A\) preferably being positive.

To convert our line equation to the standard form, we start from the slope-intercept form: \[ y = \frac{1}{4}x \] Rearranging gives: \(-\frac{1}{4}x + y = 0\). Multiplying through by 4 to remove the fraction, results in: \[ -x + 4y = 0 \] We reverse the terms to keep \(A\) positive: \[ x - 4y = 0 \] This standard form is beneficial for comparing linear features across different equations or for specific algebraic manipulations such as finding intercepts or solving systems of equations.