带有水平/垂直线的图像

Representing Horizontal and Vertical Lines with Standard Form

First, take a look at the Cartesian Coordinate system and think about this:

How do we specify a horizontal or vertical line? They are less complex than the other lines we would have, since we already know they are either parallel to the $x$-axis or the $y$-axis.

These rules are for straight lines.

Horizontal Lines

All horizontal lines are parallel to the $x$-axis. They cross (or intersect with) the $y$-axis.

To write the equation of a horizontal line, we only need to specify where it intersects with the $y$-axis. This will look like $$y=k,$$ where $k$ is where the line crosses the $y$-axis.

Can we reach this conclusion from the standard form of linear equations? The answer is yes. A line is horizontal when $\color{#8925ae}{A=0}$ in the equation ${\color{#8925ae}{A}}x+By=C$.

When $A=0$, we can rearrange the standard from equation to be: $$\begin{align} Ax+By&=C\\ ({\color{#8925ae}{0}})x+By&=C\\ By&=C\\ y&=\frac CB \end{align}$$ We can further gather that the $y$-intercept, $k$, is the same as $\frac{C}{B}$. $$y=k=\frac{C}{B}$$

The slope of a horizontal line is zero!

Vertical Lines

All vertical lines are parallel to the $y$-axis. They all cross (or intersect with) the $x$-axis.

To write the equation of a vertical line, we only need to specify where it intersects with the $x$-axis. This will look like $$x=j,$$ where $j$ is where the line crosses the $x$-axis.

We can reach this conclusion from the standard form of linear equations by letting $\color{#8925ae}{B=0}$.

When $B=0$, we can rearrange the standard from equation to be: $$\begin{align} Ax+By&=C\\ Ax+({\color{#8925ae}{0}})y&=C\\ Ax&=C\\ x&=\frac CA \end{align}$$ We can further gather that the $x$-intercept, $j$, is the same as $\frac{C}{A}$. $$x=j=\frac{C}{A}$$

The slope of a vertical line is undefined!

Recap

When using the standard form of an equation

• the line will be horizontal when $A=0$.
• the line will be vertical when $B=0$.

Solution

Vertical lines, in standard form, will have their $B$ term equal to $0$. $$Ax+0y=C$$ We can see that the second answer choice, $5x+0y=15$, is of this form.

Whenever we simplify this form, we get: $$Ax=C\quad\Rightarrow\quad x=\frac CA$$ The third answer, $x=0$, is of this form. Here the fraction $\frac CA=0$.

(Video) Standard Form of Horizontal and Vertical Lines

by Sarah Messing

This video by Sarah Messing shows how to find the standard form of a horizontal or vertical line given a point.

$\large{\text{Summary}}$

Example 1

Suppose we want to write the standard form of the equation of the horizontal line through $(6,-5)$.

To start, we can plot the given point, $(6,-5)$, and draw a horizontal line through it.

Notice that the $y$-coordinate at any point on this line is always $-5$. It's a constant. So we can write: $$y=-5$$ This is the same as writing it as $0x+y=-5$ but we just simplify and remove the $0x$. So, $y=-5$ is in standard form.

Example 2

Suppose now we want to write the equation in standard form of the vertical line that passes through $(-2,-7)$.

Graphing this gives us:

This time the $x$-coordinate is always constant. At any point on the line, the $x$-coordinate is $-2$. So we say, $$x=-2$$ We could write this as $x+0y=-2$, but this is more simplified.

In General

The standard form for an equation is: $$Ax+By=C$$

Let's plug in some values.

Any horizontal line will have $A=0$. $$\begin{align} 0x+By&=C\\ By&=C\\ y&=\frac CB \end{align}$$ This $\frac CB$ is just the $y$-intercept.

Any vertical line will have $B=0$. $$\begin{align} Ax+0y&=C\\ Ax&=C\\ x&=\frac CA \end{align}$$ This $\frac CA$ is just the $x$-intercept.

Horizontal Line Equations

Horizontal lines have varying $x$-values, but a constant $y$-value. In other words, the $y$-value is not dependent on any $x$-value...it is constant! Horizontal lines have equations that look like this: $$y=a$$ where $a$ is a real number. In this equation, $a$ is the fixed value that $y$ must take, despite the varying $x$-values.

Vertical Line Equations

Vertical lines have varying $y$-values, but a constant $x$-value. Vertical lines have equations that look like this: $$x=b$$ where $b$ is a real number. In this equation, $b$ is the fixed value that $x$ must take, despite the varying $y$-values.