বিন্দু্ু-ঢাল আকার: সমীকরণ লেখা

(Video) Writing Equations of Lines

by mathman1024

This video by mathman1024 shows you how to find the equation of a line, given two points, by using the point slope equation.

$\large{\text{Summary}}$

The equation for point slope form is: $$y-y_0=m(x-x_0)$$ To demonstrate, he finds the equation of the line that passes through the points $(-11,-2)$ and $(1,1)$. As always, the first step to solve this equation is to find the slope using the slope formula. $$\begin{align} m &=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}\\ m &=\frac{1-(-2)}{1-(-11)}\\ m &=\frac{3}{12}\\ m &=\frac14 \end{align}$$

Now that we have the slope, we use that and the coordinates for one point and plug them into the equation for $m$, $x_0$, and $y_0$. $$\begin{align} y-y_0&=m(x-x_0)\\ y-1&=\frac14(x-1)\\ \end{align}$$

And there you have it! You can also convert this to slope-intercept form by moving some terms around: $$\begin{align} y-1&=\frac14x-\frac14\\ y-1\,\color{#8925ae}{+\,1}&=\frac14x-\frac14\,\color{#8925ae}{+\,1}\\ y&=\frac14x+\frac34 \end{align}$$

Solution

First, find the slope: $$\begin{align} m &=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}\\ m &=\frac{9-6}{8-2}\\ m &=\frac{3}{6}\\ m &=\frac12 \end{align}$$

Then, plug the slope and one set of coordinates into the point slope equation. $$\begin{align} y-y_0&=m(x-x_0)\\ y-9&=\frac12(x-8)\\ \end{align}$$

Point Slope Form Equation

The equation of a line is in point slope form when it looks like: $$y-y_1=m(x-x_1)$$ Here, $x$ and $y$ are variables. They differ from $x_1$ and $y_1$ which are the coordinates of a known point on the line. Lastly, $m$ is the slope.

To find the equation of a line in this form, we either need to know the line's slope and one point on the line, or two points on the line.

Example 1: Known Slope and Point

Suppose we want to find the equation of a line in point slope form that passes through the point $(2,1)$ and has a slope of $m=\frac34$.

To solve this, we just plug these into the equation. Remember that the known point $(2,1)$ corresponds to $(x_1,y_1)$ in the equation. $$\begin{align} y-y_1&=m(x-x_1)\\ y-1&=\frac34(x-2) \end{align}$$ From here, we can leave the equation as is, or simplify it into slope intercept form. $$\begin{align} y-1&=\frac34(m-2)\\ y-1&=\frac34m-\frac64\\ y-1&=\frac34m-\frac32\\ y&=\frac34m-\frac32+1\\ y&=\frac34m-\frac12 \end{align}$$

Example 2: Two Known Points

If we only know two points of the line, we first have to find the slope.

Suppose we want to find the equation of a line in point slope form that passes through the points $(1,3)$ and $(6,8)$.

First, we use the two points to find the slope. $$\begin{align} m&=\frac{y_2-y_1}{x_2-x_1}\\ m&=\frac{8-3}{6-1}\\ m&=\frac55\\ m&=1 \end{align}$$

Then, we use the slope and one of the points to find the equation. Either point will work. $$\begin{align} y-y_1&=m(x-x_1)\\ y-8&=1(x-6)\\ y-8&=x-6 \end{align}$$

Solution

We're given a point the line passes through, $(-4,2)$, and the slope, $m=10$.

To find the equation of the line, plug these into the equation for $(x_1,y_1)$ and $m$ respectively. $$\begin{align} y-y_1&=m(x-x_1)\\ y-2&=10\left(x-(-4)\right)\\ y-2&=10(x+4) \end{align}$$

Definition of point slope form

Point slope form uses slope and a single point to describe a line. Equations take the form $y - y_1 = m(x - x_1)$, where $x$ and $y$ are variables and $m$, $y_1$, and $x_1$ are constants.

An example of an equation in this form is $y - 3 = -3(x - 0)$. In this case, the slope of the line is $3$ and a the point the line runs through is $(0, 3)$.

Here is what the graph of this equation would look like:

Note that if you didn't know the slope of the line but did know two points on it, you could also construct an equation in point slope form by using the two points to find the slope, and then choosing one set of coordinates to insert into the $y_1$ and $x_1$ spots.