91Ó°ÊÓ

The slope of a line tells us how steep the line is. It is a measure of how much the y-coordinate of a point on the line increases or decreases as the x-coordinate increases by one unit. For any two given points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

This formula is essential because it helps us characterize the direction and steepness of a line.

• If the slope \(m\) is positive, the line moves upward as it goes from left to right.
• If the slope \(m\) is negative, the line moves downward as it goes from left to right.
• If the slope \(m\) is zero, the line is horizontal.
• If the slope \(m\) is undefined (division by zero occurs), the line is vertical.

In our given exercise, we found the slope using the points \((\frac{3}{4}, \frac{8}{3})\) and \((\frac{2}{5}, \frac{2}{3})\):
Substitute the values in:
\[ m = \frac{\frac{2}{3} - \frac{8}{3}}{\frac{2}{5} - \frac{3}{4}} \]
Further simplifying:
\[ m = \frac{\frac{-6}{3}}{\frac{-7}{20}} = \frac{6}{7} \]

So, the slope of the line is \(\frac{6}{7}\). This tells us how the y-coordinates change for each unit increase in the x-coordinate.

The point-slope form of a linear equation is useful when you know one point on the line and the slope of the line. The general formula for point-slope form is:
\[ y - y_1 = m(x - x_1) \]
Here, \((x_1, y_1)\) is a known point on the line, and \(m\) is the slope. This form is very handy as it immediately gives us a way to write the equation once we know the slope and one point.

In our exercise, we used the point \((\frac{3}{4}, \frac{8}{3})\) and the slope \(m = \frac{6}{7}\):
Substituting these into our point-slope form:
\[ y - \frac{8}{3} = \frac{6}{7}(x - \frac{3}{4}) \]
Next, we distribute the slope \(\frac{6}{7}\) to simplify the equation:
\[ y - \frac{8}{3} = \frac{6}{7}x - \frac{18}{28} \]
Reduce \frac{18}{28}\ to \frac{9}{14}\:
\[ y - \frac{8}{3} = \frac{6}{7}x - \frac{9}{14} \]
This represents the line's equation in point-slope form. This step serves as the intermediate transformation before converting it to the final standard form.

The standard form of a linear equation is another way to represent a line. The standard form looks like this:
\[ Ax + By = C \]
Here, A, B, and C are integers, and typically, A should be a positive integer. The standard form is useful for quickly determining the x- and y-intercepts of the line.

To convert the point-slope form we derived earlier into the standard form, we first need to eliminate the fractions by finding a common multiple.

• We had: \( y - \frac{8}{3} = \frac{6}{7}x - \frac{9}{14} \)
• Multiply both sides by the least common denominator (LCD) of 3, 7, and 14, which is 42:
  \( 42(y - \frac{8}{3}) = 42(\frac{6}{7}x - \frac{9}{14}) \).

This gives us:
\( 42y - 112 = 36x - 27 \).

Next, we rearrange the equation to get all terms involving variables on one side and the constant on the other side:
\[ 42y - 112 = 36x - 27 \]
Combine like terms:
\( -36x + 42y = 85 \)
To ensure the equation is in the standard form \(Ax + By = C\), we move terms around to get:
\( 36x - 42y = -85 \)

This is the final equation of our line in the standard form. One last check confirms that the coefficients have no common factors, making the equation \( 36x - 42y = -85 \) the simplest form.