6.4
Standard Form

6.4 – Standard Form
 REVIEW: Slope-intercept form of a linear equation is y = mx + b

6.4 – Standard Form
 Standard Form of an equation:
Ax + By = C
 where A, B, & C are REAL #s and A & B are not 0. This is a good form to graph an equation QUICKLY.

Ax + By = C
 Rules for Standard Form:
 No fractions
 A is not negative (it can be zero, but it
CANNOT be negative).
 By the way, "integer" means no fractions, no decimals . Just clean whole numbers (or their negatives).

6.4 – Standard Form
 Using Standard Form, you can find the x-intercept and y-intercept, and then graph the equation.
 To find the intercepts, you substitute 0 for both the x and y.

6.4 – Standard Form
Example: Find the x and y intercepts for
3x + 4y = 24.
 To find the x intercept, substitute 0 for the y.
3x + 4(0) = 24
3x = 24 x = 8
 So, when y = 0, x = 8 . Your x-int. is (8, 0).

6.4 – Standard Form
 To find the y-int. substitute 0 for the x
3(0) + 4y = 24
4y = 24 y = 6
 So your y-int is (0, 6)
 Using your x and y-int, you can now graph the equation.

6.4 – Standard Form y
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1 2 3 4 5 x

6.4 – Standard Form
 Example: Graph the equation
4x + 6y = 2 y
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
3
2
1
5
4
1 2 3 4 5 x

6.4 – Standard Form
 Using Standard Form, you can write equations for vertical and horizontal lines. (You can’t write vertical lines in slope-intercept form)

6.4 – Standard Form
 Example: Graph the following y= -2 x = 4
 When graphing these, say draw a line through the ___ - axis at the number.
y y
5
4
3
2
1
5
4
3
2
1
–5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1 2 3 4 5 x –5 –4 –3 –2 –1
–1
–2
–3
–4
–5
1 2 3 4 5 x

6.4 – Standard Form
 If we are looking for the intercepts of an equation, standard from is the easiest to use. Therefore, we may want to change slope-intercept equations to standard form.

6.4 – Standard Form
Example:
 Write in Standard Form.
3 x
 First we must move the x to the other get:

2 side. So we add to both sides to
3

2
3 x
 y

6

6.4 – Standard Form
 Now we must make the numbers whole.
So we must multiply by 3 to get rid of the fraction.
3





2 x
2
3
 x

3 y y


18
6





y

4

3
4 x
 y

4

3
4
( x

2 )
4(y + 4) = 3(x – 2)
4y + 16 = 3x – 6)
4y = 3x – 22
4y – 3x= – 22
– 3x + 4y = – 22
3x – 4y = 22
Given
Multiply by 4 to get rid of the fraction.
Distributive property
Subtract 16 from both sides
Subtract 3x from both sides
Format x before y
Multiply by -1 in order to get a positive coefficient for x.

Ex. 3: Write the standard form of an equation of the line passing through (5, 4), -2/3 y

4
 
2
3
( x

5 )
3(y - 4) = -2(x – 5)
3y – 12 = -2x +10
3y = -2x +22
3y + 2x= 22
2x + 3y = 22
Given
Multiply by 3 to get rid of the fraction.
Distributive property
Add 12 to both sides
Add 2x to both sides
Format x before y

Ex. 4: Write the standard form of an equation of the line passing through (-6, -3), -1/2 y

3
 
1
2
( x

6 )
2(y +3) = -1(x +6)
2y + 6 = -1x – 6
2y = -1x – 12
2y + 1x= -12 x + 2y = -12
Given
Multiply by 2 to get rid of the fraction
Distributive property
Subtract 6 from both sides
Subtract 1x from both sides
Format x before y

Ex. 6: Write the standard form of an equation of the line passing through (5, 4), (6, 3) m
 y
2

 y
1
First find slope of the line.
x
2 x
1 y m


3

4
6


4
5


1 (

1 x
1


5 )

1
Substitute values and solve for m.
Put into point-slope form for conversion into
Standard Form Ax + By = C y – 4 = -1x + 5
Distributive property y = -1x + 9 y + x = 9
Add 4 to both sides.
Add 1x to both sides x + y = 9
Standard form requires x come before y.

Ex. 7: Write the standard form of an equation of the line passing through (-5, 1), (6, -2) m
 y
2 x
2

 y x
1
1
First find slope of the line.
m


2

1
6

(

5 )


3
6

5


3
11
Substitute values and solve for m.
y

1
 
3
11
( x

5 )
Put into point-slope form for conversion into
Standard Form Ax + By = C
11( y – 1) = -3(x + 5)
Multiply by 11 to get rid of fraction
11 y – 11 = -3x – 15
Distributive property
11 y = -3x – 4
Add 4 to both sides.
11 y + 3x = -4
Add 1x to both sides
3x + 11y = -4
Standard form requires x come before y.

6.4 – Standard Form
Writing equations in the REAL WORLD:
 You are working two jobs during the summer. You are mowing lawns and delivering newspapers. You make
$12/hour mowing lawns and $5/hour delivering newspapers . If you made a total of $130 , write an equation in standard form.
12x + 5y = 130

6.4 – Standard Form
Example #2:
 You are training to participate in the annual Ironman Championship in Kona,
Hawaii. You need to burn a total of 500 calories per day to get in proper shape.
Write an equation in standard form to find the minutes you would need to workout each day if you were to just swim and run.

6.4 – Standard Form
Activity
Bicycling
Running
Hiking
Swimming
Walking
Rowing
Calories burned per minute
10
11
7
12
2
10