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Standard Form of Linear Equations

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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 ?

Answers

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.

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