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 \((8,2)\)

Short answer

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

Step-by-step solution

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 \(\frac{1}{4}.\)

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 \((4, 1)\):\[ y - 1 = \frac{1}{4}(x - 4). \]

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:\[ y = \frac{1}{4}x. \]

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:\[ -x + 4y = 0. \]

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.