1. Functions and Equations of Two Variables Lesson 6.1

2. Functions of Two Variables Consider a function with two inputs and one output Two independent variables One dependant variable z = f ( x, y ) 7 5 43 f (x, y)

3. Example Consider the volume of a cylinder Given r = 5, h = 10 V = π *25*10 = 250π Calculator can define such functions r h

4. 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 in 3

5. 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

6. 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)

7. Solving Systems of Equations Graphical solution Solve each equation for y Graph the resulting function Note their intersections

8. 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 equation2x + 0 = 10

9. Try It Out Given Part of class determine graphical solution Part of class determine symbolic solution by substitution

10. Using Calculator Calculator can be used to solve systems of equations Use solve command Note use of and curly brackets { }

11. 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?

12. 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?

13. Try It Out Given the system Part of class do graphically Part of class do symbolically

14. Assignment Lesson 6.1 Page 460 Exercises 1 – 75 EOO