Slope-Intercept Form

You can always manipulate a linear equation to display it in various ways, but often the most useful form for interpreting a linear relationship is "slope-intercept form." Slope-intercept form is when the equation takes the format of y = mx + b, where m gives us the slope of the line and b is the y-intercept. For example, the equation y = 3x + 2 is in slope-intercept form, where the slope is 3 and the y-intercept is 2. This means that whenever x increases by 1, y increases by 3, and that the line crosses the y-axis at y=2.

To convert a linear equation from standard form (ax + by = c) to slope-intercept form (y = mx + b), you need to isolate y on one side of the equation. For example, to convert the equation 2x + 3y = 6 to slope-intercept form, you would first subtract 2x from both sides of the equation to get 3y = -2x + 6. Then, divide every term by 3 to get y = -2/3x + 2. Now that it's in slope-intercept form, we can see that the slope is -2/3 and the y-intercept is 2.

To get the slope-intercept form of a line from 2 points, you first need to find the slope of the line. The slope is found by taking the difference in the y-coordinates of the 2 points, divided by the difference in the x-coordinates of the 2 points. For example, if the 2 points are (1, 2) and (3, 6), the slope is (6-2)/(3-1) = 4/2 = 2. Once you have the slope, you can use the point-slope form of the equation of a line, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is one of the points on the line. For example, if the slope is 2 and one of the points is (1, 2), the equation of the line is y - 2 = 2(x - 1), which can be modified to y - 2 = 2x - 2, which converts to y = 2x - 0, or y = 2x. This is now in slope-intercept form, where the slope is 2 and the y-intercept is 0 (i.e. it crosses through the origin).