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# Standard Form of Linear Equations

## Explore equations in ax+by = c form

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Standard Form

At a football game, Brian is selling concessions. He sells soda for $2 apiece and popcorn for$3 per bag. At the end of the night, he has sold out of soda and has made a total of \$264. If he sold 60 sodas, how many bags of popcorn did he sell?

### Standard Form

Slope-intercept form is one way to write the equation of a line. Another way is called standard form. Standard form looks like \begin{align*}Ax + By = C,\end{align*} where \begin{align*}A, B,\end{align*} and \begin{align*}C\end{align*} are all real numbers.

Let's find the equation of a line, in standard form, where the slope is \begin{align*}\frac{3}{4}\end{align*} and passes through (4, -1).

To find the equation in standard form, you need to determine what \begin{align*}A, B,\end{align*} and \begin{align*}C\end{align*} are. Let’s start this problem by finding the equation in slope-intercept form.

\begin{align*}-1 &= \frac{3}{4}(4) + b\\ -1 &= 3 + b \\ -4 &= b\end{align*}

In slope-intercept form, the equation is \begin{align*}y = \frac{3}{4}x-4\end{align*}.

To change this to standard form we need to subtract the \begin{align*}x-\end{align*}term from both sides of the equation.

\begin{align*}- \frac{3}{4}x+y = -4\end{align*}

Let's find the slope and \begin{align*}y-\end{align*}intercept if the equation of a line is \begin{align*}5x-2y = 12\end{align*}.

To find the slope and \begin{align*}y-\end{align*}intercept of a line in standard form, we need to switch it to slope-intercept form. This means, we need to solve the equation for \begin{align*}y\end{align*}.

\begin{align*}5x - 2y &= 12\\ -2y &= -5x + 12 \\ y &= \frac{5}{2}x - 6\end{align*}

From this, the slope is \begin{align*}\frac{5}{2}\end{align*} and the \begin{align*}y-\end{align*}intercept is (0, -6).

Now, let's find the equation of the line, in standard form, using the graph below.

Here, we are given the intercepts. The slope triangle is drawn by the axes, \begin{align*}\frac{-6}{-2} = 3\end{align*}. And, the \begin{align*}y-\end{align*}intercept is (0, 6). The equation of the line, in slope-intercept form, is \begin{align*}y = 3x + 6\end{align*}. To change the equation to standard form, subtract the \begin{align*}x-\end{align*}term to move it over to the other side.

\begin{align*}-3x + y = 6 \ or \ 3x - y = -6\end{align*}

Let's find the intercepts if the equation of a line is \begin{align*}6x - 5y = 45\end{align*}.

For the \begin{align*}x-\end{align*}intercept, the \begin{align*}y-\end{align*}value is zero. Plug in zero for \begin{align*}y\end{align*} and solve for \begin{align*}x\end{align*}.

\begin{align*}6x - 5y &= 45\\ 6x - 5(0) &= 45\\ 6x &= 45 \\ x &= \frac{45}{6} \ or \ \frac{15}{2}\end{align*}

The \begin{align*}x-\end{align*}intercept is \begin{align*}\left(\frac{15}{2}, 0 \right)\end{align*}.

For the \begin{align*}y-\end{align*}intercept, the \begin{align*}x-\end{align*}value is zero. Plug in zero for \begin{align*}x\end{align*} and solve for \begin{align*}y\end{align*}.

\begin{align*}6x -5y &= 45\\ 6(0) - 5y &= 45\\ 5y &= 45 \\ y &= 9\end{align*}

The \begin{align*}y-\end{align*}intercept is (0, 9).

### Examples

#### Example 1

Earlier, you were asked to find the number of bags of popcorn Brian sold.

This type of problem is easier to write in standard form. We will call x the number of sodas and y the number of bags of popcorn. The coefficients are the cost of each item in dollars. The expression will equal the total amount of concessions Brian sold.

\begin{align*}2x+3y=264\end{align*}

Now, we know that \begin{align*}x=60\end{align*}. Plug that in and solve for y.

\begin{align*}2(60)+3y&=264 \\ 120+3y&=264\\ 3y&=144\\ y&=48\end{align*}

Therefore, Brian sold 48 bags of popcorn.

#### Example 2

Find the equation of the line, in standard form, that passes through (8, -1) and (-4, 2).

We need to first find the equation of this line in \begin{align*}y-\end{align*}intercept form and then change it to standard form. First, find the slope.

\begin{align*}\frac{2-(-1)}{-4-8} = \frac{3}{-12} = - \frac{1}{4}\end{align*}

Find the \begin{align*}y-\end{align*}intercept using slope-intercept form.

\begin{align*}2 &= - \frac{1}{4}(-4) + b\\ 2 &= 1 + b \\ 1 &= b\end{align*}

The equation of the line is \begin{align*}y = - \frac{1}{4}x + 1\end{align*}.

To change this equation into standard form, add the \begin{align*}x-\end{align*}term to both sides.

\begin{align*}& \quad \frac{1}{4}x + y = 1\end{align*}

#### Example 3

Change \begin{align*}2x + 3y = 9\end{align*} to slope-intercept form.

To change \begin{align*}2x + 3y = 9\end{align*} into slope-intercept form, solve for \begin{align*}y\end{align*}.

\begin{align*}2x + 3y &=9\\ 3y &= -2x + 9\\ y &= - \frac{2}{3}x + 3\end{align*}

#### Example 4

What are the intercepts of \begin{align*}3x - 4y = -24\end{align*}?

First, plug in zero for \begin{align*}y\end{align*} and solve for \begin{align*}x\end{align*}.

\begin{align*}3x - 4(0) &= -24\\ 3x &= -24 \\ x &= -8\end{align*}

\begin{align*}x-\end{align*}intercept is (-8, 0)

Now, start over and plug in zero for \begin{align*}x\end{align*} and solve for \begin{align*}y\end{align*}.

\begin{align*}3(0) - 4y &=-24\\ -4y &= -24 \\ y &= 6\end{align*}

\begin{align*}y-\end{align*}intercept is (6, 0)

### Review

Change the following equations into standard form.

1. \begin{align*}y = - \frac{2}{3}x + 4\end{align*}
2. \begin{align*}y = x - 5\end{align*}
3. \begin{align*}y = \frac{1}{5}x - 1\end{align*}

Change the following equations into slope-intercept form.

1. \begin{align*}4x + 5y = 20\end{align*}
2. \begin{align*}x - 2y = 9\end{align*}
3. \begin{align*}2x -3y = 15\end{align*}

Find the \begin{align*}x\end{align*} and \begin{align*}y-\end{align*}intercepts of the following equations.

1. \begin{align*}3x + 4y = 12\end{align*}
2. \begin{align*}6x - y = 8\end{align*}
3. \begin{align*}3x + 8y = -16\end{align*}

Find the equation of the lines below, in standard form.

1. slope = 2 and passes through (3, -5)
2. slope \begin{align*}= - \frac{1}{2}\end{align*} and passes through (6, -3).
3. passes through (5, -7) and (-1, 2)
4. passes through (-5, -5) and (5, -3)
5. Change \begin{align*}Ax + By = C\end{align*} into slope-intercept form.
6. From #16, what are the slope and \begin{align*}y-\end{align*}intercept equal to (in terms of \begin{align*}A, B,\end{align*} and/or \begin{align*}C\end{align*})?
7. Using #16 and #17, find one possible combination of \begin{align*}A, B,\end{align*} and \begin{align*}C\end{align*} for \begin{align*}y = \frac{1}{2}x - 4\end{align*}. Write your answer in standard form.
8. The measure of a road’s slope is called the grade. The grade of a road is measured in a percentage, for how many vertical feet the road rises or declines over 100 feet. For example, a road with a grade incline of 5% means that for every 100 horizontal feet the road rises 5 vertical feet. What is the slope of a road with a grade decline of 8%?
9. The population of a small town in northern California gradually increases by about 50 people a year. In 2010, the population was 8500 people. Write an equation for the population of this city and find its estimated population in 2017.

To see the Review answers, open this PDF file and look for section 2.3.

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### Vocabulary Language: English

Standard Form

The standard form of a line is $Ax + By = C$, where $A, B,$ and $C$ are real numbers.