Line Equation
Point-slope and Slope-intercept forms of a line
These are two common formulations for the equation for a line.
Slope-intercept form of a line is:
y=mx+b
It is useful if you want to know the slope, m, and, well, the y-intercept of the line, b. Conversely, you can find the equation for this line if you know the slope and y-intercept.
Point-slope form of a line is:
y-y1=m(x-x1)
where (x1,y1) is a point on the line.
This version of the equation for a line is useful if you know the slope of the line, and a point on the line, or conversely, you can derive this equation if you know slope and a point.
How to derive Point-slope:
slope is defined as rise over run, or change in y divided by change in x
slope = m = change in y = y-y1
change in x x-x1
So you get
m = y-y1
x-x1
If you multiply both sides by (x-x1), you end up with the point-slope formula for a line:
y-y1 = m(x-x1)
Note that you can convert between the two forms.
Point-slope to slope intercept: just solve for y, and simplify the other side.
y-y1 = m(x-x1)
y=y1+ m(x-x1)
y=y1+ mx -mx1
y=mx +(y1-mx1)
y=mx + b
So that the y-intercept is y1-mx1.
To go from slope-intercept to point-slope, just plug in a number for x, and find the corresponding y value. Plug these in for x1 and y1, and use the point-slope formula.
These are two common formulations for the equation for a line.
Slope-intercept form of a line is:
y=mx+b
It is useful if you want to know the slope, m, and, well, the y-intercept of the line, b. Conversely, you can find the equation for this line if you know the slope and y-intercept.
Point-slope form of a line is:
y-y1=m(x-x1)
where (x1,y1) is a point on the line.
This version of the equation for a line is useful if you know the slope of the line, and a point on the line, or conversely, you can derive this equation if you know slope and a point.
How to derive Point-slope:
slope is defined as rise over run, or change in y divided by change in x
slope = m = change in y = y-y1
change in x x-x1
So you get
m = y-y1
x-x1
If you multiply both sides by (x-x1), you end up with the point-slope formula for a line:
y-y1 = m(x-x1)
Note that you can convert between the two forms.
Point-slope to slope intercept: just solve for y, and simplify the other side.
y-y1 = m(x-x1)
y=y1+ m(x-x1)
y=y1+ mx -mx1
y=mx +(y1-mx1)
y=mx + b
So that the y-intercept is y1-mx1.
To go from slope-intercept to point-slope, just plug in a number for x, and find the corresponding y value. Plug these in for x1 and y1, and use the point-slope formula.
