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Transcript
Linear Functions
6.6 : General Form of the Equation
for a Linear Relation
Today’s Objectives
• Relate linear relations expressed in: slopeintercept form, general form, and slope-point
form to their graphs, including:
• Express a linear relation in different forms, and
compare graphs
• Rewrite a linear relation in either slope-intercept or
general form
• Graph, with or without technology, a linear relation in
slope-intercept, general, or slope-point form
• Identify equivalent linear relations from a set of linear
relations
• Match a set of linear relations to their graphs
Equations of a Linear Function
• We can make an equation that describes a line’s
location on a graph. This is called a linear
equation. There are three forms of linear
equation that we will be looking at:
• Standard Form: Ax + By + C = 0, where A, B, and
C are integers.
• Slope y-intercept form: y = mx + b, where m is
the slope, and b is the y-intercept.
• Slope-point form: y – y1 = m(x – x1), where m is
the slope, and the line passes through a point
located at (x1, y1)
General form (or standard form)
• Another form for the equation of a linear
function is general form, or standard form:
• 𝑨𝒙 + 𝑩𝒚 + 𝑪 = 𝟎
• In certain situations you will be asked to change
the equation from slope y-intercept form into
general form, and vice versa.
• When converting the equation into standard
form, we must remember that A, B, and C, MUST
be integers, which means all fractions need to
be removed. We also need to remember to
move all terms to the same side of the equation
to make it equal to zero.
Example
Write the following in general form:
y – 1 = 2/3(x + 2)
Multiply both sides by 3: 3(y – 1) = 2(x+2)
Expand:
3y – 3 = 2x + 4
Make equation = 0
Collect Like Terms
Arrange in proper order
3y – 2x – 7 = 0
-2x + 3y – 7 = 0
Example
Write the following in general form:
y + 2 = 3/5(x - 4)
Multiply both sides by 5
5y + 10 = 3(x – 4)
Remove the brackets
5y + 10 = 3x – 12
Collect all terms on one side of the equation
5y = 3x – 12 – 10
0 = 3x – 5y - 22
What does the graph look like?
A=0
Ax + By + C = 0
By + C = 0
y = -C/B
Since –C/B is a constant, the graph will be a horizontal line
B=0
Ax + By + C = 0
Ax + C = 0
x = -C/A
Since –C/A is a constant, the graph will be a vertical line
Example
Graph the line 2x – 5y – 10 = 0
Determine the x-intercept
Let y=0 and solve for x
2x – 5(0) – 10 = 0
2x – 10 = 0
2x = 10
x=5
(5,0)
Determine the y-intercept
Let x=0 and solve for y
2(0) – 5y – 10 = 0
-5y – 10 = 0
-5y = 10
y = -2
(0,-2)
Plot the points and draw a line through them.
Example
Determine the slope of 2x – 5y – 10 = 0
Rewrite the equation in slope-interval form:
2x – 5y – 10 = 0
-5y = -2x + 10
y = -2x + 10
-5
y = -2x/-5 + 10/-5
y = 2/5x – 2
m = 2/5
solve for y
Example
Determine the slope of -3x – 2y – 16 = 0
Rewrite the equation in slope-interval form:
-3x – 2y – 16 = 0
-2y = 3x + 16
y = 3x + 16
-2
y = 3x/-2 + 16/-2
y = -3/2x – 8
m = -3/2
solve for y