Download Different Forms of Linear Equations

Different Forms of Linear Equations
Standard Form of a Linear Equation
 If A, B, and C are real numbers, the equation Ax + By = C is called the
standard form of the equation of a line.
 It is best to write the equation with A, B, and C as integers, and A ≥ 0.
Example:
-3x + y = 4 can be expressed as 3x – y = -4 ← multiply each term by (-1)
x + 2y = 3 can be expressed as 2x + 6y = 9 ← multiply each term by (3)
Slope-Intercept Form of a Linear Equation
 The equation y = mx + b is the slope-intercept form of the equation of a
line.
 The y-intercept of the line is (0, b), and the slope of the line is m.
 The standard form of an equation of a line can be re-written in the slopeintercept form as follows:
Ax + By = C
→
By = -Ax + C
→

The slope of Ax + By = C is
-

The y-intercept of Ax + By = C
is
y=-
→ (0, )
+
Example:
-2x – 3y = 12
y=
so… – 3y = -2x +12
so… y =
so…
-4
 The slope of the line is
and the
y-intercept is (0, -4)
Rise
(0, -4)
Run
Graphing a Line Using the Slope and the y-Intercept
Step 1: Write the equation in SLOPE INTERCEPT FORM by solving for y
Step 2: Identify the y-intercept (0, b), and graph this point
Step 3: Graph a second point using the slope, starting at the y-intercept
Step 4: Draw a line connecting the two points to obtain the graph
Example:
Graph 3x + 2y = 12 by using the slope and y-intercept
Solution:
Step 1:
3x + 2y = 12
so… 2y = -3x + 12
Step 2:
The y-intercept is (0, 6): mark this point
Step 3:
The slope is m =
so… y =
+6
(0, 6)
=
(2, 3)
From (0, 6), go down 3 units
and to the right 2 units, to
obtain the point (2, 3).
Step 4:
Draw a line through the points
(0, 6) and (2, 3).
Graphing a Line Using the Slope and a Point
Step 1: Locate and graph the given point.
Step 2: Graph another point using the slope, counting from the first point
Step 3: Draw a line connecting the two points to obtain the graph
Example:
Graph the line through (-2, -4) with slope 3.
Solution:
The slope is 3, so from the point (-2, -4) go up 3 units, and to the right 1 unit to
get the point (-1, -1).
(-1, -1)
(-2, -4)
Writing an Equation of a Line Using a Slope and a Point
 By substituting given values for a slope and a point of a line into
y = mx + b, the line’s equation can be found!
Example:
Write the equation of the line with slope 2 that runs through (-4, 1) in slope
intercept-form.
**y = mx + b**
Solution:
The point (-4, 1) gives us an x-value of -4 and a y-value of 1.
So,
y = mx + b
1 = 2(-4) + b
1 = -8 + b
b=9
Therefore the equation of the line is… y = 2x + 9
Point-Slope Form of a Linear Equation
 The equation y – y1 = m(x – x1) is the point-slope equation of a line.
 The given point is (x1, y1) and the slope of the line is m
 This formula comes from rearranging the definition of the slope,
m=
Example:
Write the equation of a line with slope 2 that passes through (-4, 1) in
Slope-intercept form.
Solution:
y – y1 = m(x – x1)
y – 1 = 2(x – (-4))
y – 1 = 2(x +4)
y – 1 = 2x +8
y = 2x +9
Example:
Write the equation of a line with slope that passes through (3, -2) in
Standard form. **Ax + By = C**
Solution:
y – y1 = m(x – x1)
y – (-2) = (x – 3)
y + 2 = (x – 3)
5(y + 2) = 4(x – 3)
5y + 10 = 4x – 12
4x – 5y = 22