书写方程式

What is Standard Form?

Standard form is just another way to write a linear equation equation along with slope intercept form and point slope form.

Standard form appears in the form:

The constants, $A$, $B$, and $C$, must be integers. And $A$ must be positive.

An example of a line in standard form would be: $$4x+7y=12$$ Here, $4$, $7$, and $12$ are all whole numbers, and $4$ is positive.

Example 1

Turn $-\frac23x-\frac52y=4$ into standard form.

First, we need to remove the fractions by multiplying each term by the LCM. Here, the LCM of $2$ and $3$ is $6$. $$\begin{align} -\frac23x-\frac52y&=4\\ {\color{#8925ae}{6\,\times}}\,\left(-\frac23x-\frac52y\right)&={\color{#8925ae}{6\,\times}}\,\left(4\right)\\ -4x-15y&=24 \end{align}$$ Then, since our $A$ term is negative, we multiply everything by $-1$ to make it positive. $$\begin{align} -4x-15y&=24\\ {\color{#8925ae}{-1\,\times}}\,\left(-4x-15y\right)&={\color{#8925ae}{-1\,\times}}\,\left(24\right)\\ 4x+15y&=-24 \end{align}$$ This is now in standard form.

Always keep in mind: whatever you do to one side of the equation, you have to do to the other. Always following the Multiplication Property of Equality.

Why Use This?

In my opinion, it's the least important of the three. But one thing it's very good at is finding the $x$- and $y$-intercepts of a line.

The $x$-intercept occurs when $y=0$. Similarly, the $y$-intercept occurs when $x=0$. So to find these, you just need to plug in $x=0$ and solve for $y$, then plug in $y=0$ and solve for $x$.

Example 2

Suppose we want to graph the following equation by plotting its $x$- and $y$-intercepts. $$2x+5y=30$$ Let's start by finding the $x$-intercept. To find this, we set $y=0$. $$\begin{align} 2x+5y&=30\\ 2x+5({\color{#8925ae}{0}})&=30\\ 2x&=30\\ \frac{2x}{\color{#8925ae}2}&=\frac{30}{\color{#8925ae}2}\\ x&=15 \end{align}$$ So the $x$-intercept occurs at the point $(15,0)$.

Next, to find the $y$-intercept, we set $x=0$. $$\begin{align} 2x+5y&=30\\ 2({\color{#8925ae}{0}})+5y&=30\\ 5y&=30\\ \frac{5y}{\color{#8925ae}5}&=\frac{30}{\color{#8925ae}5}\\ y&=6 \end{align}$$ The $y$-intercept occurs at the point $(0,6)$.

Plotting these points, we can draw a line through them to get:

Conversion

Convert from slope intercept form to standard form and back is easy. Let's look at some examples.

Example 3

Put the following equation into standard form: $$y=\frac43x+8$$

Start by getting $x$ and $y$ together on one side of the equation. $$\begin{align} y&=\frac43x+8\\ y\,{\color{#8925ae}{-\,\frac43x}}&=\frac43x\,{\color{#8925ae}{-\,\frac43x}}+8\\ -\frac43x+y&=8 \end{align}$$

Next, remove all fractions by multiplying all terms by the LCM. In this example, it's $3$. $$\begin{align} -\frac43x+y&=8\\ {\color{#8925ae}{3\,\times}}\,\left(-\frac43x+y\right)&={\color{#8925ae}{3\,\times}}(8)\\ -4x+3y&=24 \end{align}$$

Finally, multiply all terms by $-1$ so the $A$ term becomes positive. $$\begin{align} -4x+3y&=24\\ {\color{#8925ae}{-1\,\times}}\,\left(-4x+3y\right)&={\color{#8925ae}{-1\,\times}}(24)\\ 4x-3y&=-24 \end{align}$$

Example 4

Put the following equation in slope-intercept form: $$3x+5y=2$$

Start by putting $x$ on the opposite side of $y$. $$\begin{align} 3x+5y&=2\\ 3x\,{\color{#8925ae}{-\,3x}}+5y&=2\,{\color{#8925ae}{-\,3x}}\\ 5y&=2-3x \end{align}$$

Then divide to isolate the $y$. $$\begin{align} 5y&=2-3x\\ \frac{5y}{\color{#8925ae}5}&=\frac{2-3x}{\color{#8925ae}5}\\ y&=\frac25-\frac35x \end{align}$$

Solution

1. $3x+7y=12\quad$This is in standard form. There are no fractions and the $A$ term, $3$, is nonnegative.
2. $4y-9x=18\quad$At first glance this looks like standard form but it is not. The $x$ term is negative, which we cannot have.
3. $2x-7y=\frac12\quad$This is not in standard form. The $C$ term, $\frac12$, is a fraction. All terms must be integers.
4. $x+y=-100\quad$This is in standard form. There are no fractions and the $A$ term, $1$, is positive. (The $1$ is not shown because it is in front of the $x$. So, $1x=x$.)

(Video) Finding the Equation of a Line in Standard Form

by Mahalo Math

This video by Mahalo Math explains how to convert an equation in slope-intercept form to standard form.

$\large{\text{Summary}}$

Equations in standard form follow the structure: $$Ax + By = C$$ $A, B$, and $C$ are all integers. $A$ should also always be positive.

To get an equation into standard form, just follow these steps:

1. Isolate the constant (the term with no variable) on the right side of the equation by just adding and subtracting terms from both sides.
2. If there are any fractions, multiply the whole equation by the lowest common denominator.
3. If the constant in the $A$ position is negative, multiply the whole equation by $-1$.

Example

Suppose we want to find the standard form equivalent to: $$y=\frac73 x-12$$

First, we want to isolate the constant, $-12$. To do this, we can subtract $\frac73x$ from both sides of the equation. Then rearrange so the $x$ term is first. $$\begin{align} y&=\frac73x-12\\ y\,{\color{#8925ae}{-\,\frac73x}}&=\frac73x\,{\color{#8925ae}{-\,\frac73x}}-12\\ y-\frac73x&=-12\\ -\frac73x+y&=-12 \end{align}$$ Next, we remove fractions by multiplying by the lowest common denominator, $3$. $$\begin{align} -\frac73x+y&=-12\\ {\color{#8925ae}{3\,\times}}\,\left(-\frac73x+y\right)&={\color{#8925ae}{3\,\times}}\,\left(-12\right)\\ -7x+3y&=-36\\ \end{align}$$

Finally, since our $x$ term's coefficient is negative, we multiply the equation by $-1$ to make it positive. $$\begin{align} -7x+3y&=-36\\ {\color{#8925ae}{-1\,\times}}\,\left(-7x+3y\right)&={\color{#8925ae}{-1\,\times}}\,\left(-36\right)\\ 7x-3y&=36 \end{align}$$ In this form, $A=7$, $B=-3$, and $C=36$.

These two equations are equivalent but are in different forms. $$\begin{array}{ccc} \text{Slope Intercept}&\rightarrow&\text{Standard Form}\\ y=\frac73x-12&\rightarrow&7x-3y=36 \end{array}$$

Solution

First, we want to get the constant, $15$, alone on one side. $$\begin{align} y&=\frac38x+15\\ y\,{\color{#8925ae}{-\,\frac38x}}&=\frac38x\,{\color{#8925ae}{-\,\frac38x}}+15\\ y-\frac38x&=15\\ -\frac38x+y&=15 \end{align}$$

Next, we get rid of fractions by multiplying by the lowest common denominator, $8$. $$\begin{align} -\frac38x+y&=15\\ {\color{#8925ae}{8\,\times}}\,\left(-\frac38x+y\right)&={\color{#8925ae}{8\,\times}}\,\left(15\right)\\ -3x+8y&=120 \end{align}$$

Finally, since our $A$ coefficient is negative, we need to multiply the entire equation by $-1$ to make it positive. $$\begin{align} -3x+8y&=120\\ {\color{#8925ae}{-1\,\times}}\,\left(-3x+8y\right)&={\color{#8925ae}{-1\,\times}}\,\left(120\right)\\ 3x-8y&=-120 \end{align}$$

Translation between standard form and slope-intercept form $\\$

A line in standard form looks like:

A line in slope-intercept form looks like:

The process of translating between the two might seem confusing at first, just practice a few problems and you will get a better handle of it. Make sure you are not memorizing anything. Instead, understand how you transform and manipulate the equation to fit either of the forms above.

From standard form to slope-intercept $\\$

From slope-intercept to standard form $\\$

Example $\\$

A couple of quick facts about standard form:

• No fractions: We don't usually see fractions, making the equation "pretty"
• $A$ is positive: note that this does not mean the slope is positive. This is just a convention of standard form.
• Easy to find $x$- and $y$-intercepts: when a line is in standard form, we can find $x$-intercept and $y$-intercept easier.