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Functions and Equations of Two Variables

- Lesson 6.1

Functions of Two Variables

- Consider a function with two inputs and one

output - Two independentvariables
- One dependant variable
- z f ( x, y )

7

5

f (x, y)

43

Example

- Consider the volume of a cylinder
- Given r 5, h 10
- V p 2510 250p
- Calculator can define such functions

r

h

Solving for One of the Variables

- How high must the cylinder be for
- Radius of 6 inches
- Volume of 230 cubic inches
- Write out the formula
- Substitute in the known quantities
- Solve for the unknown value

6

h

230 in3

Linear Equation in Two Variables

- Format
- Where a, b, and k are constants
- This can also be thought of as a function in two

variables - Example
- Now note that there are many (x, y) ordered pairs

that can be considered solutions

System of Equations

- If we have two equations in two variables it is

possible that one ordered pair is the solution

for both equations - Which of the following ordered pairs are

solutions for the system? (3, 2) (3, -4)

(5, 0)

Solving Systems of Equations

- Graphical solution
- Solve each equation for y
- Graph the resulting function
- Note their intersections

Symbolic Solution

- Solve one of the equations for one of the

variables - x y 5
- Substitute the expression in for that variable in

the other equation - 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

- Given
- Part of class determine graphical solution
- Part of class determine symbolic solution by

substitution

Using Calculator

- Calculator can be used to solve systems of

equations - Use solve command
- Note use of
- and
- curly brackets

Systems of Non-Linear Equations

- Consider
- Note one of equations is not linear
- Same types of solutions can often be used
- Symbolic by substitution
- Graphical
- What kind of graphs are demonstrated?

Number of Solutions

- System of linear equations
- One solution no solutions many

solutions - For non linear systems
- Depends on the type of the graphs involved
- What different possibilities exist for a line and

a parabola?

Try It Out

- Given the system
- Part of class do graphically
- Part of class do symbolically

Assignment

- Lesson 6.1
- Page 460
- Exercises 1 75 EOO

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