The "point-slope" form of the equation of a straight line is:y − y1 = m(x − x1) Show The equation is useful when we know:
and want to find other points on the line. Have a play with it first (move the point, try different slopes): Now let's discover more. What does it stand for?(x1, y1) is a known point m is the slope of the line (x, y) is any other point on the line Making sense of itIt is based on the slope: Slope m = change in y change in x = y − y1 x − x1
So, it is just the slope formula in a different way! Now let us see how to use it.Example 1:slope "m" = 31 = 3 y − y1 = m(x − x1) We know m, and also know that (x1, y1) = (3,2), and so we have: y − 2 = 3(x − 3) That is a perfectly good answer, but we can simplify it a little: y − 2 = 3x − 9 y = 3x − 9 + 2 y = 3x − 7 Example 2:m = −3 1 = −3 y − y1 = m(x − x1) We can pick any point for (x1, y1), so let's choose (0,0), and we have: y − 0 = −3(x − 0) Which can be simplified to: y = −3x Example 3: Vertical LineWhat is the equation for a vertical line? In fact, this is a special case, and we use a different equation, like this: x = 1.5 Every point on the line has x coordinate 1.5, What About y = mx + b ?You may already be familiar with the "y=mx+b" form (called the slope-intercept form of the equation of a line). It is the same equation, in a different form! The "b" value (called the y-intercept) is where the line crosses the y-axis. So point (x1, y1) is actually at (0, b) and the equation becomes: Start withy − y1 = m(x − x1) (x1, y1) is actually (0, b):y − b = m(x − 0) Which is:y − b = mx Put b on other side:y = mx + b This calculator will find the equation of a line (in the slope-intercept, point-slope, and general forms) given two points or the slope and one point, with steps shown. Related calculators: Slope Calculator, Parallel and Perpendicular Line Calculator SolutionYour input: find the equation of a line given two points $$$P=\left(-4, 7\right)$$$ and $$$Q=\left(1, 2\right)$$$. The slope of a line passing through the two points `P=(x_1, y_1)` and `Q=(x_2, y_2)` is given by `m=(y_2-y_1)/(x_2-x_1)`. We have that $$$x_1=-4$$$, $$$y_1=7$$$, $$$x_2=1$$$, $$$y_2=2$$$. Plug the given values into the formula for slope: $$$m=\frac{\left(2\right)-\left(7\right)}{\left(1\right)-\left(-4\right)}=\frac{-5}{5}=-1$$$. Now, the y-intercept is `b=y_1-m*x_1` (or `b=y_2-m*x_2`, the result is the same). $$$b=7-\left(-1\right) \cdot \left(-4\right)=3$$$. Finally, the equation of the line can be written in the form `y=mx+b`. $$$y=-x+3$$$. Answer: The slope of the line is $$$m=-1$$$. The equation of the line in the slope-intercept form is $$$y=-x+3$$$. The equation of the line in the point-slope form is $$$y - 7 = - (x + 4)$$$. The equation of the line in the point-slope form is $$$y - 2 = - (x - 1)$$$. The general equation of the line is $$$x + y - 3 = 0$$$. |