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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 34 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.

114=34(4)+b=3+b=b

In slope-intercept form, the equation is y=34x4 .

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

34x+y=4

Example B

The equation of a line is 5x2y=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 .

5x2y2yy=12=5x+12=52x6

From this, the slope is 52 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, 62=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 3xy=6

Example D

The equation of a line is 6x5y=45 . What are the intercepts?

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

6x5y6x5(0)6xx=45=45=45=456 or 152

The x intercept is (152,0) .

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

6x5y6(0)5y5yy=45=45=45=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 coefficients 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)+3y120+3y3yy=264=264=144=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 3x4y=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.

2(1)48=312=14

Find the y intercept using slope-intercept form.

221=14(4)+b=1+b=b

The equation of the line is y=14x+1 .

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

14x+y=1

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

2x+3y3yy=9=2x+9=23x+3

3. Copy Example D to find the intercepts of 3x4y=24 . First, plug in zero for y and solve for x .

3x4(0)3xx=24=24=8

x intercept is (-8, 0)

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

3(0)4y4yy=24=24=6

y intercept is (6, 0)

Explore More

Change the following equations into standard form.

  1. y=23x+4
  2. y=x5
  3. y=15x1

Change the following equations into slope-intercept form.

  1. 4x+5y=20
  2. x2y=9
  3. 2x3y=15

Find the x and y intercepts of the following equations.

  1. 3x+4y=12
  2. 6xy=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 =12 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=12x4 . 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

Standard Form

Standard Form

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

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