Problem 41 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the general form of the equation of the line that passes through the two points. \((-1,5),(7,3)\)

Step-by-Step Solution

Answer

The general form of the equation of the line is \(x + 4y = 21\).

Step 1: Finding the Slope

Calculate the slope (m) of the line using the formula \(m = (y2 - y1)/(x2 - x1)\). Substituting \(x1 = -1\), \(y1 = 5\), \(x2 = 7\), \(y2 = 3\) you get \(m = (3 - 5)/(7 - (-1)) = -2/8 = -1/4\)

Step 2: Converting to Point-Slope Form

The point-slope form of a line is given by \(y - y1 = m(x - x1)\), where m is the slope and (x1, y1) is a point on the line. Substituting \(m = -1/4\), \(x1 = -1\), and \(y1 = 5\) into the formula, we get \(y - 5 = -1/4(x - (-1)) => y - 5 = -1/4x - 1/4\)

Step 3: Converting to General Form

Multiply every term by 4 to get rid of the denominator in the term -1/4x. This results in \(4y - 20 = -x - 1\). Rearrangiging this we get \(x + 4y = 21\) as the general form of the equation.

Key Concepts

Slope Calculation

Understanding the slope of a line is crucial in geometry as it represents how steep a line is. The slope is typically denoted as 'm' and is calculated by comparing the vertical change (rise) to the horizontal change (run) between two points on a line.

Here's a simple way to find the slope when you have two points: Take the difference in the y-values and divide it by the difference in the x-values. The formula is expressed as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

For example, with the points \((-1,5),(7,3)\), we subtract the y-values (3 - 5) and x-values (7 - (-1)) and get the slope \( m = \frac{3 - 5}{7 - (-1)} = -\frac{1}{4} \). Remember, a negative slope indicates that the line is decreasing, while a positive slope indicates an increasing line.

If the slope is zero, the line is horizontal and if the slope is undefined (division by zero), the line is vertical.

Point-Slope Form

Point-slope form is an equation of a straight line that uses the slope and a specific point on the line. This form is particularly useful when you already know these two pieces of information.

The formula looks like this: \( y - y_1 = m(x - x_1) \) where 'm' is the slope and \((x_1, y_1)\) is the point on the line used to create the formula.

Using the example at hand, with a known slope \(m = -\frac{1}{4}\) and the point \((-1, 5)\), we plug these into the formula yielding \(y - 5 = -\frac{1}{4}(x - (-1))\), which simplifies to \(y - 5 = -\frac{1}{4}x - \frac{1}{4}\). This equation allows us to see how y-values change with x, given the slope and a fixed point on the line.

General Form of a Line

The general form of a line's equation is typically presented as \(Ax + By = C\), where 'A', 'B', and 'C' are integers and 'A' should be non-negative. This form is helpful for analyzing lines using classic algebraic methods.

To convert an equation from point-slope form to the general form, we reorganize and simplify the equation so that x and y terms are on one side, and the constant is on the other. Here's a quick guide to make that conversion:

• Multiply each term to eliminate any fractions.
• Rearrange the terms to get them on one side, often aiming to move the term with x to the front.
• If necessary, simplify terms further by combining like-terms or reducing numbers to their lowest terms.

Continuing from our previous point-slope form example, \(y - 5 = -\frac{1}{4}x - \frac{1}{4}\), after multiplication and rearrangement, we arrive at \(x + 4y = 20 + 1\), which simplifies to the final general form \(x + 4y = 21\).