Finding the y-Intercept (b) of a Line in Slope-Intercept Form: A Step-by-Step Guide with Example

When working with linear equations, the slope-intercept form is a powerful tool. This form, y mx b, provides a clear view of the line’s slope and y-intercept. In this guide, we will explore how to find the value of b, the y-intercept, when given two points on a line. We will use an example with points A(6, 6) and B(12, 3).

Understanding the Slope-Intercept Form

The slope-intercept form of a line is y mx b, where:

m represents the slope of the line. b represents the y-intercept of the line, the point where the line crosses the y-axis.

Calculating the Slope (m)

To find the slope m, we use the formula:

m (y? - y?) / (x? - x?)

Given the points A(6, 6) and B(12, 3), we can substitute the values into the formula:

y? 3 y? 6 x? 12 x? 6

m (3 - 6) / (12 - 6) -3 / 6 -1 / 6

Using the Point-Slope Form to Find the Y-Intercept (b)

Once we have the slope, we can use the point-slope form to find the y-intercept b. The point-slope form is y - y? m(x - x?). Using point A(6, 6) and the slope m -1 / 6, we get:

y - 6 -1 / 6(x - 6)

To convert this into the slope-intercept form, we solve for y:

y -1 / 6x -1 / 6(-6) 6

Now, simplify:

y -1 / 6x 1 6 -1 / 6x 7

Therefore, the y-intercept b is 7, but let's verify using the given values:

Substitute point A(6, 6) into y -1 / 6x b:

6 -1 / 6(6) b

6 -1 b

b 6 1 5

Verification of the Equation

Let's verify the equation y -1 / 6x 5 with the given points:

For point A(6, 6):

y -1 / 6(6) 5 -1 5 6 (Checks out)

For point B(12, 3):

y -1 / 6(12) 5 -2 5 3 (Checks out)

Conclusion

In this guide, we successfully found the y-intercept (b) of the line using the slope-intercept form. The steps involved calculating the slope, converting the point-slope form to the slope-intercept form, and verifying the result with the given points. The final equation of the line is:

y -1 / 6x 5

This approach ensures that you can accurately determine the y-intercept of any line, given two points.