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2.3: Standard Form

Difficulty Level: At Grade Created by: CK-12
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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?


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*}Ax+By=C, where \begin{align*}A, B,\end{align*}A,B, and \begin{align*}C\end{align*}C are all integers. In the Review Queue above, the equations from problems 1 and 2 are in standard form. Once they are solved for \begin{align*}y,\end{align*}y, they will be in slope-intercept form.

Example A

Find the equation of a line, in standard form, where the slope is \begin{align*}\frac{3}{4}\end{align*}34 and passes through (4, -1).

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

\begin{align*}-1 &= \frac{3}{4}(4) + b\\ -1 &= 3 + b \\ -4 &= b\end{align*}114=34(4)+b=3+b=b

In slope-intercept form, the equation is \begin{align*}y = \frac{3}{4}x-4\end{align*}y=34x4.

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

\begin{align*}- \frac{3}{4}x+y = -4\end{align*}34x+y=4

However, we are not done. In the definition, \begin{align*}A, B,\end{align*}A,B, and \begin{align*}C\end{align*}C are all integers. At the moment, \begin{align*}A\end{align*}A is a fraction. To undo the fraction, we must multiply all the terms by the denominator, 4. We also will multiply by a negative so that the \begin{align*}x-\end{align*}xcoefficient will be positive.

\begin{align*}& -4 \left(-\frac{3}{4}x+y = -4 \right)\\ & \qquad \ \ 3x - 4y = 16\end{align*}4(34x+y=4)  3x4y=16

Example B

The equation of a line is \begin{align*}5x-2y = 12\end{align*}5x2y=12. What are the slope and \begin{align*}y-\end{align*}yintercept?

Solution: To find the slope and \begin{align*}y-\end{align*}yintercept 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*}y.

\begin{align*}5x - 2y &= 12\\ -2y &= -5x + 12 \\ y &= \frac{5}{2}x - 6\end{align*}5x2y2yy=12=5x+12=52x6

From this, the slope is \begin{align*}\frac{5}{2}\end{align*}52 and the \begin{align*}y-\end{align*}yintercept is (0, -6).

Example C

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

Solution: Here, we are given the intercepts. The slope triangle is drawn by the axes, \begin{align*}\frac{-6}{-2} = 3\end{align*}62=3. And, the \begin{align*}y-\end{align*}yintercept is (0, 6). The equation of the line, in slope-intercept form, is \begin{align*}y = 3x + 6\end{align*}y=3x+6. To change the equation to standard form, subtract the \begin{align*}x-\end{align*}xterm to move it over to the other side.

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

Example D

The equation of a line is \begin{align*}6x - 5y = 45\end{align*}6x5y=45. What are the intercepts?

Solution: For the \begin{align*}x-\end{align*}xintercept, the \begin{align*}y-\end{align*}yvalue is zero. Plug in zero for \begin{align*}y\end{align*}y 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).

Intro Problem Revisit 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 coefficents are the cost of each item in dollars. The expression will equal the total amount of concessions Brian sold.


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.

Guided Practice

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

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

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


1. Like with Example A, 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 and multiply by 4 to get rid of the fraction.

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

2. 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*}

3. Copy Example D to find 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)


Standard Form (of a line)
When a line is in the form \begin{align*}Ax + By = C\end{align*} and \begin{align*}A, B,\end{align*} and \begin{align*}C\end{align*} are integers.


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.

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

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Date Created:
Mar 12, 2013
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