Transcript Functions and Equations of Two Variables Lesson 6.1 Functions of Two Variables  Consider a function with two inputs and one output   Two independent variables One dependant variable z = f.

Functions and
Equations of Two
Variables
Lesson 6.1
Functions of Two Variables
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Consider a function with two inputs and one
output
5
7
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Two independent
variables
One dependant
variable
z = f ( x, y )
f (x, y)
43
Example
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V  f (r, h)    r  h
2
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h
Given r = 5, h = 10
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r
Consider the volume of a cylinder
V = π *25*10 = 250π
Calculator can define such functions
Solving for One of the
Variables
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6
How high must the cylinder be for
Radius of 6 inches
Volume of 230 cubic inches
Write out the formula
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Substitute in the known quantities
Solve for the unknown value
230    36  h
230
h
36
h
230 in3
Linear Equation in Two
Variables
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Format
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a x b y  k
Where a, b, and k are constants
This can also be thought of as a function in two
variables
Example
6 x  3 y  24
f ( x, y )  24
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Now note that there are many (x, y) ordered
pairs that can be considered solutions
System of Equations
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If we have two equations in two variables it is
possible that one ordered pair is the solution for
both equations
x y 5
2 x  y  10
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Which of the following ordered pairs are
solutions for the system?
(3, 2) (3, -4) (5, 0)
Solving Systems of Equations
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Graphical solution
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Solve each equation for y
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Graph the resulting function
Note their intersections
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Symbolic Solution
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Solve one of the equations
for one of the variables
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2 x  y  10
x=y+5
Substitute the expression in for that variable in
the other equation
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x y 5
2 ( y + 5 ) + y = 10
Result is an equation in one variable
Solve that equation for the variable 3y = 0
Substitute that value back into the other
equation
2x + 0 = 10
Try It Out
7x  2 y  5
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Given
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Part of class determine graphical solution
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Part of class determine symbolic solution by
substitution
x  9 y  10
Using Calculator
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Calculator can be used to solve systems of
equations
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Use solve command
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Note use of
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and
curly brackets {
}
Systems of Non-Linear
Equations
x  y 1
2
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Consider
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Note one of equations is not linear
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Same types of solutions can often be used
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3x  y  1
Symbolic by substitution
Graphical
What kind of graphs are demonstrated?
Number of Solutions
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System of linear equations
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One solution
no solutions
many solutions
For non linear systems
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Depends on the type of the graphs involved
What different possibilities exist for a line and a
parabola?
Try It Out
x  y 1
2
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Given the system
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Part of class do graphically
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Part of class do symbolically
3x  y  1
Assignment
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Lesson 6.1
Page 460
Exercises 1 – 75 EOO