How to Write the Equation of a Line in Slope-Intercept Form Through Given Points

Understanding Slope-Intercept Form

The slope-intercept form of a line is a convenient way to express a linear equation because it directly reveals two key pieces of information: the slope of the line and the y-intercept. There are two main forms of the slope-intercept equation: y mx c, where c is the y-intercept, and y mx - d, where d is the x-intercept. However, the standard form we often use is the y mx b, which is equivalent to the first form where b represents the y-intercept.

Given Points: (2,1) and (6,-5)

Given two points, (2,1) and (6,-5), we can determine the equation of the line in slope-intercept form. Let's start by calculating the slope (m) of the line.

Step 1: Calculate the Slope

The formula for calculating the slope between two points (x1, y1) and (x2, y2) is:

[ m frac{y_2 - y_1}{x_2 - x_1} ]

Substituting the given points into the formula:

[ m frac{-5 - 1}{6 - 2} frac{-6}{4} -frac{3}{2} ]

Step 2: Determine the y-Intercept

The y-intercept is the value of y when x 0. We can determine it using the point-slope form and the point (2,1) where x 2 and y 1:

[ y - y_1 m(x - x_1) ]

Substituting the slope and the point (2,1):

[ y - 1 -frac{3}{2}(x - 2) ]

Solving for y when x 0:

[ 1 -frac{3}{2}(2) c ]

[ 1 -3 c ]

[ c 4 ]

Step 3: Write the Equation in Slope-Intercept Form

Now that we have the slope (m -(frac{3}{2})) and the y-intercept (c 4), we can write the equation of the line:

[ y -frac{3}{2}x 4 ]

Conclusion

In summary, the equation of the line passing through points (2,1) and (6,-5) in slope-intercept form is: