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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?

### Guidance

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

#### Example A

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

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

$-1 &= \frac{3}{4}(4) + b\\-1 &= 3 + b \\-4 &= b$

In slope-intercept form, the equation is $y = \frac{3}{4}x-4$ .

To change this to standard form we need to subtract the $x-$ term from both sides of the equation.

$- \frac{3}{4}x+y = -4$

#### Example B

The equation of a line is $5x-2y = 12$ . What are the slope and $y-$ intercept?

Solution: To find the slope and $y-$ 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 $y$ .

$5x - 2y &= 12\\-2y &= -5x + 12 \\y &= \frac{5}{2}x - 6$

From this, the slope is $\frac{5}{2}$ and the $y-$ intercept 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, $\frac{-6}{-2} = 3$ . And, the $y-$ intercept is (0, 6). The equation of the line, in slope-intercept form, is $y = 3x + 6$ . To change the equation to standard form, subtract the $x-$ term to move it over to the other side.

$-3x + y = 6 \ or \ 3x - y = -6$

#### Example D

The equation of a line is $6x - 5y = 45$ . What are the intercepts?

Solution: For the $x-$ intercept, the $y-$ value is zero. Plug in zero for $y$ and solve for $x$ .

$6x - 5y &= 45\\6x - 5(0) &= 45\\6x &= 45 \\x &= \frac{45}{6} \ or \ \frac{15}{2}$

The $x-$ intercept is $\left(\frac{15}{2}, 0 \right)$ .

For the $y-$ intercept, the $x-$ value is zero. Plug in zero for $x$ and solve for $y$ .

$6x -5y &= 45\\6(0) - 5y &= 45\\5y &= 45 \\y &= 9$

The $y-$ 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.

$2x+3y=264$

Now, we know that $x=60$ . Plug that in and solve for y.

$2(60)+3y&=264 \\120+3y&=264\\3y&=144\\y&=48$

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 $2x + 3y = 9$ to slope-intercept form.

3. What are the intercepts of $3x - 4y = -24$ ?

1. Like with Example A, we need to first find the equation of this line in $y-$ intercept form and then change it to standard form. First, find the slope.

$\frac{2-(-1)}{-4-8} = \frac{3}{-12} = - \frac{1}{4}$

Find the $y-$ intercept using slope-intercept form.

$2 &= - \frac{1}{4}(-4) + b\\2 &= 1 + b \\1 &= b$

The equation of the line is $y = - \frac{1}{4}x + 1$ .

To change this equation into standard form, add the $x-$ term to both sides.

$& \quad \frac{1}{4}x + y = 1$

2. To change $2x + 3y = 9$ into slope-intercept form, solve for $y$ .

$2x + 3y &=9\\3y &= -2x + 9\\y &= - \frac{2}{3}x + 3$

3. Copy Example D to find the intercepts of $3x - 4y = -24$ . First, plug in zero for $y$ and solve for $x$ .

$3x - 4(0) &= -24\\3x &= -24 \\x &= -8$

$x-$ intercept is (-8, 0)

Now, start over and plug in zero for $x$ and solve for $y$ .

$3(0) - 4y &=-24\\-4y &= -24 \\y &= 6$

$y-$ intercept is (6, 0)

### Vocabulary

Standard Form (of a line)
When a line is in the form $Ax + By = C$ and $A, B,$ and $C$ are real numbers.

### Practice

Change the following equations into standard form.

1. $y = - \frac{2}{3}x + 4$
2. $y = x - 5$
3. $y = \frac{1}{5}x - 1$

Change the following equations into slope-intercept form.

1. $4x + 5y = 20$
2. $x - 2y = 9$
3. $2x -3y = 15$

Find the $x$ and $y-$ intercepts of the following equations.

1. $3x + 4y = 12$
2. $6x - y = 8$
3. $3x + 8y = -16$

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

1. slope = 2 and passes through (3, -5)
2. slope $= - \frac{1}{2}$ and passes through (6, -3).
3. passes through (5, -7) and (-1, 2)
4. passes through (-5, -5) and (5, -3)
5. Change $Ax + By = C$ into slope-intercept form.
6. From #16, what are the slope and $y-$ intercept equal to (in terms of $A, B,$ and/or $C$ )?
7. Using #16 and #17, find one possible combination of $A, B,$ and $C$ for $y = \frac{1}{2}x - 4$ . 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.

### Vocabulary Language: English

Standard Form

Standard Form

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