Another special way to write linear
equations is called
standard form:
Ax + By = C
Another special way to write linear
equations is called
standard form:
Ax + By = C
A, B, and C are usually written as
integers, and A is usually not less
than zero.
Another special way to write linear
equations is called
standard form:
Ax + By = C
A and B cannot both be zero, and
A, B, and C, are usually relatively
prime.
Ax + By = C
y = -3x + 4
Ax + By = C
x
+3
x
+3
y = -3x + 4
Ax + By = C
x
+3
x
+3
y = -3x + 4
Ax + By = C
x
+3
x
+3
y = -3x + 4
3x + y = 4
Ax + By = C
x
+3
x
+3
y = -3x + 4
3x + y = 4
A = 3, B = 1, C = 4
Ax + By = C
y=
2
3x
–3
Ax + By = C
y=
2
3x
–3
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
-23 x + y = -3
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
2 x + y = -3 • 3
3 3
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
2 x + y = -3 • 3
3 3
-2x + 3y = -9
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
2 x + y = -3 • 3
3 3
-1 -2x + 3y = -9 • -1
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
2 x + y = -3 • 3
3 3
-1 -2x + 3y = -9 • -1
2x – 3y = 9
Ax + By = C
2x
2x
+ 3 2 + 3
y = 3x – 3
2 x + y = -3 • 3
3 3
-1 -2x + 3y = -9 • -1
2x – 3y = 9
A = 2, B = -3, C = 9
1
x
6
+
2
3
=
3
4y
12
(
1
x
6
+
)
2
3
=
3
4y•
12
12
(
1
x
6
12 1 x
1 • 6
)
+
2
3
+
12
1
3
4y•
=
•
2
3
12
= 34 y • 12
1
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
2x + 8 = 9y
1
3
12
1
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
-9y + -8 + 2x + 8 = 9y + -9y + -8
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
-9y + -8 + 2x + 8 = 9y + -9y + -8
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
-9y + -8 + 2x + 8 = 9y+ -9y + -8
-9y + 2x = -8
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
-9y + -8 + 2x + 8 = 9y+ -9y + -8
-9y + 2x = -8
2x – 9y = -8
12
2
(
1
x
6
12 1 x
1 • 6
1
)
+
2
3
+
4
12
1
=
•
3
4y•
2
3
1
12
= 34 y •
1
3
12
1
-9y + -8 + 2x + 8 = 9y+ -9y + -8
-9y + 2x = -8
2x – 9y = -8
A = 2, B = -9, C = -8
A horizontal line such as
y = 42
is written both in standard form
(0x + y = 42)
and slope-intercept form
(y = 0x + 42)
A = 0, B = 1, C = 42
m = 0, b = 42
A vertical line such as
x = 23
is written in standard form
(x + 0y = 23)
A = 1, B = 0, C = 23
Equations in standard form are
usually written in "lowest terms."
Equations in standard form are
usually written in "lowest terms."
This is another way of saying that
A, B, and C are relatively prime.
Equations in standard form are
usually written in "lowest terms."
6x + 12y = 18
Equations in standard form are
usually written in "lowest terms."
6x + 12y = 18
would be changed to the equivalent
equation:
x + 2y = 3
Equations in standard form are
usually written in "lowest terms."
9x – 12y = -24
Equations in standard form are
usually written in "lowest terms."
9x – 12y = -24
would be changed to the equivalent
equation:
3x – 4y = -8
In a previous lesson we discovered
that the slope of:
5x + 7y = 24
is - 5
7
The slope of an equation written in
standard form, Ax + By = C is:
A
m=
B
In a previous lesson we discovered
that the slope of:
5x + 7y = 24
is - 5
7
A
m= B
2x + 3y = 12
m=
A
m= B
2x + 3y = 12
2
m= 3
A
m= B
2x + 3y = 12
2
m= 3
2x – 3y = 9
m=
A
m= B
2x + 3y = 12
2
m= 3
2x – 3y = 9
m=
A
m= B
2x + 3y = 12
2x – 3y = 9
2
m= 3
2
m = -3
A
m= B
2x + 3y = 12
2x – 3y = 9
2
m= 3
2 2
m = -3 = 3
A
m= B
2x – 3y = 9
2
m= 3
2
m= 3
3x + y = 4
m=
2x + 3y = 12
A
m= B
2x – 3y = 9
2
m= 3
2
m= 3
3x + 1y = 4
m=
2x + 3y = 12
A
m= B
2x + 3y = 12
2x – 3y = 9
3x + 1y = 4
2
m= 3
2
m= 3
3
m=
1
A
m= B
2x + 3y = 12
2x – 3y = 9
3x + 1y = 4
2
m= 3
2
m= 3
3
m = = -3
1
Another special way to write linear
equations is called
standard form:
Ax + By = C
The slope of an equation written in
standard form is:
A
m= B