Understanding the Equation of a Line Through Given Points Using Slope-Intercept Form

When faced with the task of finding the equation of a line that passes through specific points, the slope-intercept form is a powerful tool. The equation of a line can be written in the form y mx b, where m is the slope and b is the y-intercept. This form allows us to easily understand how the line behaves and to quickly determine its key features.

Step-by-Step Guide: Finding the Equation of a Line

Step 1: Calculate the Slope (m)

To find the slope (m) of the line, we use the formula:

Formula: m (y2 - y1) / (x2 - x1)

Let's take the points (2, -4) and (1, 6). Here, x1 2, y1 -4, x2 1, and y2 6.

Calculation:

m (6 - (-4)) / (1 - 2) (6 4) / (1 - 2) 10 / -1 -10

Step 2: Use Point-Slope Form to Find the Equation

Once we have the slope, we can use the point-slope form of the equation of a line:

Formula: y - y1 m(x - x1)

Using the point (2, -4), we can plug in the known values:

y - -4 -10(x - 2)

Simplification:

y 4 -1 20

Solving for y:

y -1 16

Step 3: Verify and Finalize the Equation

The final equation of the line is:

y -1 16

Additional Examples: Exploring Different Methods

Example 1: Slope and y-Intercept

Consider the points (3, -6) and (-2, -1). We can determine the slope (m) as follows:

Slope m (y2 - y1) / (x2 - x1) (-1 - (-6)) / (-2 - 3) (-1 6) / (-2 - 3) 5 / -5 -1

Therefore, the line can be expressed in the form y -1x b, where b is the y-intercept. Using either point, we can solve for b:

Using (3, -6): -6 -1 * 3 b rarr; b -6 3 -3

Final equation is:

y -x - 3

Example 2: Point-Gradient Form

Let's use the points (1, 6) and (-2, 3) to find the equation of the line:

Slope (m) (y2 - y1) / (x2 - x1) (3 - 6) / (-2 - 1) -3 / -3 1

Using the point-slope form:

y - y1 m(x - x1)

Using (1, 6): y - 6 1(x - 1)

Simplification:

y - 6 x - 1

Solving for y:

y x 5

General Form and Verification

Another common form to express the equation of a line is the general or standard form: Ax By C 0. For the points (2, -4) and (1, 6), we can also express the equation as:

y - 4 -10(x - 2) rarr; y - 4 -1 20

Final general form:

1 y - 24 0

Conclusion

Using these steps, we can easily find the equation of a line that passes through given points. The slope-intercept form is highly useful, but there are different methods available. By understanding these methods and practicing with various examples, you can confidently solve similar problems and gain a deeper understanding of linear equations and their graphical representations.