Question

Given the following two points, write the equation of the line in slope-intercept form, point-slope form, and standard form: (2, 3) and (-1,9). Use complete sentences to explain how you move from one form to the other.

Answer 1

The slope of the line through the points (2, 3) and (-1, 9) is -2, and using this slope, the y-intercept is calculated to be 7. Hence, the equations of the line are in slope-intercept form: y = -2x + 7, in point-slope form: y - 3 = -2(x - 2), and in standard form: 2x + y = 7.

Step-by-step explanation:

First, let's find the slope of the line using the two points (2, 3) and (-1, 9). The slope (m) is the rise over the run, which we can calculate as:

m = (y2 - y1) / (x2 - x1) = (9 - 3) / (-1 - 2) = 6 / -3 = -2

Now, using the slope-intercept form y = mx + b, we can plug in one of the points and the slope to solve for the y-intercept (b):

3 = (-2)(2) + b

b = 3 + 4 = 7

The slope-intercept form of the line is:

y = -2x + 7

To write this equation in point-slope form, we use the formula y - y1 = m(x - x1), where (x1, y1) is either one of the given points:

y - 3 = -2(x - 2)

The point-slope form is:

y - 3 = -2(x - 2)

For the standard form (Ax + By = C), we manipulate the slope-intercept form:

2x + y = 7

The standard form of the line is:

2x + y = 7