线的方程式

The equation of a line has no exponents or square roots, meaning the highest power of the equation of the line is $1$. Any equation with a power of exactly $1$ is called linear.

Recall that the slope of a line is the ratio between the vertical and horizontal distance between any two points. This value is always constant in a line, which means points on a line will have the same slope. The $y$-intercept of a line is the $y$ value at the point where the line hits the $y$-axis.

The two most common forms for writing the equation of a line are $y = mx+ b$ (slope-intercept form) and $y - y_1 = m(x - x_1)$ (point-slope form), where $m$ is the slope of the line, and $b$ is the y-intercept.

The slope of the line is sometimes called $\frac{rise}{run}$ (said: rise over run). Rise is the vertical distance between two points, while run is the horizontal distance; to get the slope, the rise and run must be between the same two points on the line, although the choice of these two points is arbitrary.

Given two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope of the line passing between the two points is $\frac{y_2 - y_1}{x_2 - x_1}$.

From here, we can either choose to put the equation into either form. The easier way would be to put it into point-slope form, as $(x_1, y_1)$ are the points you put into the equation.

For example, the equation of the line passing through the points $(4,2)$ and $(9,8)$ would have a slope of $\frac{8-2}{9-4}$, or $\frac{6}{5}$. The equation of the line would then be $y - 2 = \frac{6}{5}(x - 4)$.

However, if we wanted to put the equation into slope-intercept form, we would choose one of our coordinates, plug it into the slope-intercept form, and solve for $b$, the $y$-intercept of the line.

From our previous example, we would have: $$y=mx+b$$ $$2 = \frac{6}{5}(4) + b$$ Solving, we find that $b$ is equal to $-\frac{14}{5}$.

Thus, the equation of our line is $y = \frac{6}{5}x - \frac{14}{5}$.

Let's try some practice. You are given the input $x$ values of $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ and the equation $y-2=-2(x+5)$.

$m$ represents the slope of the equation in point slope form $y - y_1 = m(x - x_1)$. So, the slope of this equation is $-2$.

The simplest way to find this would be to simplify this equation into slope intercept form ($y=mx+b$). First, distribute the $-2$. $$y-2=-2x-10$$ Then, add $2$ to both sides. $$y=-2x-8$$ Now, we can easily see that the $y$-intercept is $\color{blue}{-8}$.

The values that are on this line have the slope of $-2$. You can check each point with a known point to make sure they have the correct slope. For example, we know the point $(\color{blue}{0},\color{green}{-8})$ is on the line. We can use that and the equation for slope to make sure the other points are on the line. Let's check the point $(12, -32)$.

$$\begin{align} m&=\frac{y_2-y_1}{x_2-x_1}\\ m&=\frac{\color{green}{-8}-y_1}{\color{blue}{0}-x_1}\\ m&=\frac{\color{green}{-8}-(-32)}{\color{blue}{0}-12}\\ m&=\frac{24}{-12}\\ m&=-2 \end{align}$$

We can see the point $(12, -32)$ is on the line $y-2=-2(x+5)$. You can also check to see if these points are on the line by plugging in the $x$ values into the equation and see if you get the correct $y$ value. For example, if we plug in $\color{red}{17}$, we see $$\begin{align} y-2&=-2(x+5)\\ y-2&=-2(\color{red}{17}+5)\\ y-2&=-2(22)\\ y-2&=-44\\ y&=-42 \end{align}$$

We see that the output for $x=17$ is $-42$. That is not what the answer choice was, so it is not correct.

You can use either of these two ways to also check the points $(-5,2)$ and $(-7, 6)$. They are both on the line $y-2=-2(x+5)$.

To find the output values, we must plug in the $x$ values into the equation $y-2=-2(x+5)$. Doing that, we see that the $y$-values are $-10, -12, -14, -16, -18, -20, -22, -24, -26,$ and $-28$.

(Video) Equations of Lines

by yaymath

In this video, yaymath shows you how to find the equation of a line.

$\large{\text{Summary}}$

Slope-intercept form

To find the equation of a line you first need to find the slope, which you can do as long as you have two coordinates. The clearest form of a line is slope-intercept form, which is $y=mx+b$. In slope-intercept form, $m$ is the slope and $b$ is the $y$-intercept.

Finding the slope: Plug the coordinates into the equation $\frac{y_2 - y_1}{x_2 - x_1}$, then simplify. For more information on this, see this explanation.

Finding the $y$-intercept: $(0,b)$ The $y$-intercept occurs when $x=0$. If you don't already have a point where $x$ is $0$, you can choose any one of your points and plug it into the equation $y=mx+b$ (where $m$ is already substituted by the slope you found). Since you know the slope and have an $x$ and $y$ value, the only remaining variable is $b$—so just solve the equation, and you can find your $y$-intercept!

Point slope form

The other way to figure out the equation of a line is by using point slope form. $$y-y_0=m(x-x_0)$$

Here, you take any point already known and plug it into the corresponding $x_0$ and $y_0$ spots. Then, find the slope using the technique above and substitute it in for $m$, and simplify so the $y$ is by itself. The result is the equation of the line!

Finding the Equation of a Line

Let's first look at the standard form for the equation of a line. This form is also called slope-intercept form. (You'll see why it's given that name in a little bit). $$y=mx+b$$

Now, let's identify what each of the terms in the equation means:

$(x,y)$ = coordinate pair / point (a location on a graph with an $x$- and $y$-axis)

$m$ = slope of the line

**$b$ = $y$-intercept of the line **(where the line crosses the $y$-axis)

You can identify the $y$-intercept of a line simply by looking at it on a graph. This is the point where the line crosses the $y$-axis (and where $x=0$). Let's practice.

Remember that the equation of a line can be written as $y=mx+b$. In the equation $4x-5$, $b=-5$. This means that the $y$-intercept is $-5$.

Another way of thinking about this is looking at the value of $y$ when $x$ is $0$, since $x=0$ is exactly the equation of the $y$-axis.

We've figured out how to find the $b$ ($y$-intercept) in our $y=mx+b$ equation. Now, let's talk about how to find $m$. Recall that $m=\text{slope}$.

One way to think about the slope of a line is how steep it is. You can also think about it as $\frac{\text{rise}}{\text{run}}$, or how many steps upwards the line takes for every step to the side.

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Although this equation might look confusing at first, it's really pretty straightforward. All you need to do is pick $2$ coordinate pairs ($2$ points) on the line. That's where you get the $x$ and $y$ values from. Let's practice.

$$m=\frac{-4-6}{-5-5}$$ $$m=\frac{-10}{-10}$$ $$m=1$$

You can also determine whether the slope of a line is positive or negative by looking at a graph of the line. Here's a comparison:

(Videos) Finding the Slope and the Equation of a Line

by PatrickJMT

First, here's a video by PatrickJMT showing you how to find the slope of a line given two points. When calculating the slope, it does not matter which points you use. The formula is: $$m=\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{\text{rise}}{\text{run}}$$

$\large{\text{Summary}}$

The second video by PatrickJMT shows you how to find the equation of a line given two points. The equation is called point slope form. As long as you know any two points on a line, you can find the equation of the line. The equation is: $$y-y_0=m(x-x_0)$$ First you need to calculate the slope using the equation above. Once you find that put it in where the $m$ is. Then take any point on the line and put its $x$ and $y$ values in for $x_0$ and $y_0$ in the equation. Solve for $y$ and you'll have the equation of the line.

(Video) Slope Intercept Form

by Math Meeting

This video by Math Meeting describes the slope-intercept form of a line.

$\large{\text{Summary}}$

Slope intercept form: $y=mx+b$. $\\$ In order to get the equation of a line into slope intercept form, get "$y$" by itself.

Example: $$\begin{align} 3x+2y&=4\\ 2y&=-3x+4\\ y&=-\frac{3}{2}x+2 \end{align}$$

When graphed, the slope of this equation is $-\frac{3}{2}$ and the $y$-intercept is $2$.

(Video) Equation of a Line Given Point and Slope

by Math Meeting

This video by Math Meeting explains how to find the equation of a line given a point on the line and its slope.

$\large{\text{Summary}}$

If we have a point $(1,-3)$ and $m=4$, we can create an equation of a line.

Here are the steps:

1. Find the slope. (given)

2. Use point slope equation.

3. Solve for $y$ to make equation in slope intercept form.

Point-slope form: $y-y_1=m(x-x+1)$

Let's plug in the values of our problem. $$y+3=4(x+1)$$

We can then simplify and solve for $y$. $$y=4x-7$$