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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.

### Answers for Review Problems

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.