How to rewrite an equation in standard form

Step 1

Índice Show

Example 1: Rewriting Equations in Standard Form
Example 2: Standard Form Equations
Example 3: Eliminating Fractions

The standard form of a linear equation is .

Step 2

Step 3

Move all terms containing variables to the left side of the equation.

Subtract from both sides of the equation.

Step 4

Add to both sides of the equation.

Enter YOUR Problem

There are times when you may want to rewrite linear equations in standard form. In this video we will rewrite three equations and then graph one of them. The characteristics and advantages of standard form will also be discussed.

where $A$, $B$ and $C$ are integers, and $A$ is positive. When an equation written in standard form it is easier to find

the x and y-intercepts.

On your own:

1. Write $\color{blue}{3y-8=5x+2}$ in standard form and then graph the equation.
2. Write $\color{blue}{0=2(2x-5y+1)}$ in standard form.
3. Write $\color{blue}{y=\frac{4}{5}x-\frac{2}{3}}$ in standard form.

Solutions:

1. Original problem: $\color{#d8608c}{3y-8=5x+2}$ Remember the number in front of $x$ needs to be positive. To do that we need to multiply

each term by $-1$.

$$ \begin{align*} 3y-8\color{#d8608c}{+8} &= 5x+2\color{#d8608c}{+8} \\ 3y &= 5x+10\\ \color{#d8608c}{-5x}+3y &= 5x\color{#d8608c}{-5x}+10\\ \color{#d8608c}{-1}(-5x+3y &= 10) \\ \color{#d8608c}{5x-3y} &= \color{#d8608c}{-10} \end{align*} $$

To graph the equation we will make a table of points starting with
the x- and y-intercepts.

$$ \begin{array}{c|c} x & y\\ \hline 0 & 3\frac{1}{3}\\ \hline -2 & 0\\ \hline 1 & 5\\ \hline 4 & 10\\ \hline \end{array}

2. Original problem: $\color{#09a067}{0=2(2x-5y+1)}$
You should simplify you equation if you can. In this case all the terms are divisible by 2.

$$ \begin{align*} 0 &= \color{#09a067}{4x-10y+2}\\ \color{#09a067}{-2} &= 4x-10y\\ \\[1px] \frac{-2}{\color{#09a067}{-2}} &= \frac{4x}{\color{#09a067}{-2}}-\frac{10}{\color{#09a067}{-2}}\\ \\[1px] \color{#09a067}{-1} &= \color{#09a067}{2x-5y} \quad or \quad \color{#09a067}{2x-5y=-1} \end{align*} $$

3. Original problem: $\color{purple}{y=\frac{4}{5}x-\frac{2}{3}}$

Remember fractions are not allowed in standard form, to clear the fractions we will need to multiply each term by $15$.

$$ \begin{align*} y\color{purple}{*15} &= \frac{4}{5}x\color{purple}{*15}-\frac{2}{3}\color{purple}{*15} \\[1px] 15y &= 12x-10\\ 15y\color{purple}{+10} &= 12x-10\color{purple}{+10}\\ 15y+10 &= 12x\\ 15y\color{purple}{-15y}+10 &= 12x\color{purple}{-15y}\\ \color{purple}{10} &= \color{purple}{12x-15y} \quad or \quad \color{purple}{12x-15y=10} \end{align*}

We know that equations can be written in slope intercept form or standard form.

Let's quickly revisit standard form. Remember standard form is written:

Ax +By= C

We can pretty easily translate an equation from slope intercept form into standard form. Let's look at an example.

Example 1: Rewriting Equations in Standard Form

Rewrite y = 2x - 6 in standard form.

Standard Form: Ax + By = C

This means that we want the variables (x & y) to be on the left-hand side and the constant (6) to be on the right-hand side.

When we move terms around, we do so exactly as we do when we solve equations! So, remember... Whatever you do to one side of the equation, you must do to the other side!

Solution

That was a pretty easy example. We just need to remember that our lead coefficient should be POSITIVE!

Let's take a look at another example that involves fractions. There is one other rule that we must abide by when writing equations in standard form.

Equations that are written in standard form:

Ax + By = C

CANNOT contain fractions or decimals! A, B, and C MUST be integers!

Let's take a look at an example.

Example 2: Standard Form Equations

Rewrite y = 1/2x + 4 in standard form.

We now know that standard form equations should not contain fractions. Therefore, let's first eliminate the fractions.

Since the only fraction is is 1/2, we can multiply all terms by the denominator (2) to eliminate the fraction.

Solution

Now, let's look at an example that contains more than one fraction with different denominators.

If you find that you need more examples or more practice problems, check out the Algebra Class E-course. You'll find additional examples on video, lots of practice problems with detailed solutions and little "tips" to help you through!