Finding the Equation of a Line Through Given Coordinates

Understanding how to find the equation of a line that passes through two given points is a fundamental concept in algebra and geometry. This tutorial will explore the methods and steps to derive the equation of a line using the coordinates (-12, 76) and (76, -12). We will also derive the slope of the line and use it to find the y-intercept, ultimately writing the equation in slope-intercept form.

1. Understanding the General Form of a Linear Equation

The general form of the equation of a straight line is given by:

y mx c
Where: m is the slope of the line. c is the y-intercept, the point where the line crosses the y-axis.

2. Deriving the Slope (m)

First, let's calculate the slope m. The formula for the slope between two points (x1, y1) and (x2, y2) is:

m (frac{y2 - y1}{x2 - x1})

Using the points (7, 6) and (-2, -6) (in any order), we get:

m (frac{-6 - 6}{-2 - 7} frac{-12}{-9} frac{4}{3})

Therefore, the slope m is (frac{4}{3}).

3. Finding the y-Intercept (c)

Next, we use one of the points to find the y-intercept c. Let's use the point (7, 6). Substituting the point and the slope into the equation y mx c:

6 (frac{4}{3}) * 7 c

Solving for c:

6 (frac{28}{3}) c

c 6 - (frac{28}{3})

c (frac{18}{3}) - (frac{28}{3})

c -(frac{10}{3})

The y-intercept c is -(frac{10}{3}).

4. Writing the Equation of the Line

Now, we can write the equation of the line in the form y mx c:

y (frac{4}{3})x - (frac{10}{3})

Therefore, the equation of the line that passes through the coordinates (-12, 76) and (76, -12) is y (frac{4}{3})x - (frac{10}{3}).

5. Visual Representation of the Line

The graph of the line y (frac{4}{3})x - (frac{10}{3}) is shown in green. The points (7, 6) and (-2, -6) are marked at the intersections of the yellow and green lines, as shown in the figure. This visualization can help in understanding the geometric representation of the equation.

6. Alternative Derivation for Slope

An alternative method for finding the slope is to calculate it as follows:

m (frac{y2 - y1}{x2 - x1})

Substituting the points (7, 6) and (-2, -6) (in any order), we get:

m (frac{-6 - 6}{-2 - 7} frac{-12}{-9} frac{4}{3})

Thus, the slope is again (frac{4}{3}).

7. Summary

In summary, the slope of the line passing through the coordinates (-12, 76) and (76, -12) is (frac{4}{3}) and the y-intercept is -(frac{10}{3}). Therefore, the equation of the line is: