根据已知两点书写方程式

(Video) Writing Equations of Lines

by mathman1024

This video by mathman1024 teachs you how to find and write the equation of a line when given at least $2$ points.

$\large{\text{Summary}}$

The problem he solves is, "Find the equation of a line that passes through the points $(3,-4)$ and $(-1,2)$." To solve this problem you can either use the slope-intercept equation or the point slope equation.

Remember that the form for slope-intercept is: $$y=mx+b$$

And point slope is: $$y-y_0=m(x-x_0)$$

Finding the equation

Mathman1024 takes the steps to use slope-intercept form in this video. The first step for these problems is to find the slope $m$. We can use our points $\color{#8925AE}{(3,-4)}$ and $\color{#29A603}{(-1,2)}$ to find it. $$m=\frac{\text{rise}}{\text{run}}=\frac{y_2-y_1}{x_2-x_1}\\ \begin{align} m&=\frac{{\color{#29A603}{2}}-{\color{#8925AE}{(-4)}}}{{\color{#29A603}{-1}}-{\color{#8925AE}{3}}}\\ m&=\frac{6}{-4}\\ m&=-\frac32 \end{align}$$

Now using slope and either of the points, we can find the $y$-intercept with the slope intercept equation. $$\begin{align} y&=mx+b\\ 2&=-\frac32(-1)+b\\ 2&=\frac32+b\\ 2\,{\color{#8925AE}{-\,\frac32}}&=\frac32\,{\color{#8925AE}{-\,\frac32}}+b\\ \frac12&=b \end{align}$$

So now that we have the slope and the $y$-intercept we can find the equation of the line. $$y = mx + b \\ \\ y=-\frac32x+\frac12$$

We can use the steps above to find the slope ($m = -1$) and the $y$-intercept ($b = 6$), then just plug them into our point-slope equation: $$y = mx + b\\\ \\ \begin{align} y = -x + 6&\text{, or}\\ y = 6 - x& \end{align}$$

Let's try a more difficult one now. You are given an arithmetic sequence $a_1,a_2,\dots,a_{27}$. If $a_{\color{#29A603}{12}} = \color{blue}{6}$ and $a_{\color{purple}{27}} = \color{Orange}{36}$, find $a_{\color{red}{1}}$.

There are two ways to solve this problem.

First Way

Given that this is an arithmetic sequence, we know there is a linear relationship between $i$ and $a_i$. We know the $\color{#29A603}{12}^{\text{th}}$ element is $\color{blue}{6}$ and the $\color{#8925AE}{27}^{\text{th}}$ element is $\color{Orange}{36}$.

We can treat these as two points: $(\color{green}{12}, \color{blue}{6}), (\color{#8925AE}{27}, \color{Orange}{36})$ on some line $y=mx+b$. To find $a_1$, it is the same to find $y$ given that $x=1$.

First, let's calculate the slope: $$\frac{{\color{Orange}{36}}-\color{blue}{6}}{{\color{#8925AE}{27}}-\color{green}{12}}=\frac{30}{15}=2$$

Plug $m=2$ and point $(\color{green}{12}, \color{blue}{6})$ back into the equation: $$\begin{align} y&=mx+b\\ \color{blue}{6}&=2({\color{#29A603}{12}})+b\\ {\color{blue}{6}}-24&=b\\ -18&=b \end{align}$$

So the equation to the line is $y=2x-18$. Plug in $x=1$ and solve for $y$: $$\begin{align} y&=2x-18\\ y&=2-18\\ y&=\color{Red}{-16}\\ \end{align}$$

Second Way

We are given that $a_{\color{#29A603}{12}}=\color{blue}{6}$ and $a_{\color{#8925AE}{27}}=\color{Orange}{36}$. Using these, we can find $d$. Recall that the later term is equal to the earlier term plus some multiple of the common difference. The multiple is the later subscript minus the earlier subscript. $$a_{\text{later}}=a_{\text{earlier}}+(\text{later}-\text{earlier})d$$

In terms of our problem it will be: $$a_{\color{purple}{27}}=a_{\color{#29A603}{12}}+({\color{#8925AE}{27}}-{\color{#29A603}{12}})d\\ a_{\color{#8925AE}{27}}=a_{\color{#29A603}{12}}+15d$$

Plug in the given values and solve for $d$. $$\begin{align} a_{\color{#8925AE}{27}}&=a_{\color{#29A603}{12}}+15d\\ 36&={\color{blue}{6}}+45d\\ 30&=15d\\ 2&=d \end{align}$$

Since we want to find the first term, we can use this method again. $$a_{\color{#29A603}{12}}=a_{\color{Red}{1}}+(\color{green}{12}-1)d\\ a_{\color{#29A603}{12}}=a_{\color{Red}{1}}+11d$$

Plug in our known values and solve for $a_{\color{red}{1}}$. \begin{#29A603} a_{\color{green}{12}}&=a_{\color{Red}{1}}+11d\\ \color{blue}{6}&=a_{\color{Red}{1}}+11(2)\\ \color{blue}{6}&=a_{\color{Red}{1}}+22\\ \color{Red}{-16}&=a_{\color{Red}{1}} \end{align}

Writing Equations

Here are the steps:

1. Find the slope ($m$).
2. Find the $y$-intercept ($b$).
3. Replace $m$ and $b$ with the slope and $y$-intercept (respectively) in $y = mx + b$

In this explanation, we're going to be working with the points $(4, 5)$ and $(-1, 4)$.

So, first we find the slope. We can do that by using rise over run. If possible, you can graph the points and count like in the graph above.

We can count starting at the point $(-1,4)$. Going up one unit, and over $5$ units, we are at the point $(4,5)$. So the slope is $\frac{1}{5}$. Now, we find the $y$-intercept. Let's use the equation $y=mx+b$ and plug in $x$, $y$, and $m$. We see, $$5=\frac{1}{5}(4)+b$$

This simplifies to $b=\frac{21}{5}$. We can also draw the line through the points on the graph to see where the line hits the $y$-axis. This of course is not as accurate.

Now, we see we have the line: $$y=\frac{1}{5}x+\frac{21}{5}$$

Write a Slope-Intercept Equation from Two Known Points

To write an equation in slope intercept form from two points that you already know, you'll have to take a couple of steps:

1. Find the slope ($m$).
2. Find the $y$-intercept ($b$).
3. Replace $m$ and $b$ with the slope and $y$-intercept (respectively) in $y = mx + b$

In this explanation, we're going to be working with the points $(1, 5)$ and $(-2, -4)$.

Step 1: Find the slope

The slope of the line that goes through the two points is $3$, so $m = 3$.

Step 2: Find the $y$-Intercept

The $y$-intercept is defined as the $y$ value when $x = 0$. It's the point when the line crosses the y axis.

One way to find the $y$-intercept is to find the point slope equation ($y - y_0 = m(x - x_0)$) of your graph using one of your two points and the slope you already found. Then, just set $x$ to $0$ and solve the equation for $y$.

Work through the steps in the explanation above on your own using our two points, $(1, 5)$ and $(-2, -4)$.

The $y$-intercept should be $2$. Here are the steps I took: $$\begin{align} \text{Slope:}&\quad 3\\ \text{Coordinates:}&\quad(1, 5) \end{align}$$ $$\begin{align} y - y_0 &= m(x - x_0)\\ y - 5 &= 3(x - 1) \end{align}$$ Now, set $x$ to $0$. $$\begin{align} y - 5 &= 3(0 - 1)\\ y - 5 &= 3(-1)\\ y - 5 &= -3\\ y &= 2 \end{align}$$

Step 3: Plug the numbers into the slope intercept template

Now we know that the slope is $3$ ($m = 3$) and the $y$-intercept is $2$ ($b = 2$), so we can plug those numbers into our slope intercept template ($y = mx + b$).

The final equation is $y = 3x + 2$. Great job!

If you want to double check your answer, you can try and graph the equation you solved for!

Another way to find slope-intercept form is to find the slope and y-intercept using a graph.