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Transcript
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.
Standard Form
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
–5
–4
–3
–2
–1
–1
–2
–3
–4
–5
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
–5
–4
–3
–2
y
5
5
4
4
3
3
2
2
1
1
–1
–1
1
2
3
4
5
x
–5
–4
–3
–2
–1
–1
–2
–2
–3
–3
–4
–4
–5
–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 y  2 x  6 in Standard Form.
3
First we must move the x to the other
2
side. So we add  x to both sides to
3
get:
2
 x y 6
3
6.4 – Standard Form
Now we must make the numbers whole.
So we must multiply by 3 to get rid of
the fraction.
 2

3  x  y  6  
 3

 2 x  3 y  18
Ex. 2: Write
3
y  4  (in
x standard
2)
form.
4
3
y  4  ( x  2)
4
Given
4(y + 4) = 3(x – 2)
Multiply by 4 to get rid of the fraction.
4y + 16 = 3x – 6)
Distributive property
4y = 3x – 22
Subtract 16 from both sides
4y – 3x= – 22
Subtract 3x from both sides
– 3x + 4y = – 22
Format x before y
3x – 4y = 22
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
2
y  4   ( x  5)
3
Given
3(y - 4) = -2(x – 5)
Multiply by 3 to get rid of the fraction.
3y – 12 = -2x +10
Distributive property
3y = -2x +22
Add 12 to both sides
3y + 2x= 22
2x + 3y = 22
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
1
y 3   ( x  6)
2
Given
2(y +3) = -1(x +6)
Multiply by 2 to get rid of the fraction
2y + 6 = -1x – 6
Distributive property
2y = -1x – 12
Subtract 6 from both sides
2y + 1x= -12
x + 2y = -12
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)
y2  y1
m
x2  x1
First find slope of the line.
3  4 1
m

 1
65 1
y  4  1( x  5)
y – 4 = -1x + 5
y = -1x + 9
y+x=9
x+y=9
Substitute values and solve for m.
Put into point-slope form for conversion into
Standard Form Ax + By = C
Distributive property
Add 4 to both sides.
Add 1x to both sides
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)
y2  y1
m
x2  x1
m
First find slope of the line.
 2 1
3 3


6  (5) 6  5 11 Substitute values and solve for m.
y 1  
3
( x  5)
11
Put into point-slope form for conversion into
Standard Form Ax + By = C
– 1) = -3(x + 5)
11y – 11 = -3x – 15
11(y
11y
11y
= -3x – 4
Multiply by 11 to get rid of fraction
Distributive property
Add 4 to both sides.
+ 3x = -4
3x + 11y = -4
Add 1x to both sides
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
Calories burned
per minute
Bicycling
10
Running
11
Hiking
7
Swimming
12
Walking
2
Rowing
10