1. Chapter 8 Section 2: Equations with Two Variables. February 5 th , 2009
2. Warm Up <ul><li>Graph these points : </li></ul><ul><li>A (-1, 4) </li></ul><ul><li>B (4, 2) </li></ul><ul><li>C (-3, -2) </li></ul><ul><li>Evaluate </li></ul><ul><li>3x – 5, for x = 2 </li></ul><ul><li>2x + 3y, for x = -4 and y =5. </li></ul><ul><li>Is this Relation a Function ? </li></ul><ul><li>{(1, 2), (2, 2), (3, 1), (2, 4)} </li></ul>
3. Two Variables <ul><li>We’ve talked about equations like this: </li></ul><ul><li>2x + 5 = 7x (SADMEP & Combine Like Terms) </li></ul><ul><li>But what about equations that look like this: </li></ul><ul><li>y = 3x + 4 (???) </li></ul><ul><li>An ORDERED PAIR that will make a two variable statement true is called a Solution to the equation. </li></ul><ul><li>Two variables, x = ? and y = ?, so (?, ?) is the ordered pair that is a solution to a particular two variable equation. </li></ul>
4. Example <ul><li>Some times we’ll just give you one of the numbers in the ordered pair. </li></ul><ul><li>You plug it in, and chug out the other variable in the ordered pair. </li></ul><ul><li>Like this: y = 3 x + 4, when x = -1 . </li></ul><ul><li>y = 3 (-1) + 4 </li></ul><ul><li> y = -3 + 4 </li></ul><ul><li>y = 1 </li></ul><ul><li>The ordered pair is then, ( -1 , 1), for this equation. </li></ul>
5. Try These: Find y, and give Ordered Pair, x = -3 <ul><li>y = 2x + 1 </li></ul><ul><li>y = -4x + 3 </li></ul><ul><li>y = 0x -4 </li></ul>y = -5, (-3, -5) y = -15, (-3, 15) y = -4, (-3, -4)
6. Real Life Example <ul><li>The equation t = 21 – 0.01n models the normal low July temperature in degrees Celsius at Mt. Rushmore, South Dakota. In the equation, t is the temperature at n meters above the base of the mountain. Find the normal low July temperature at 300m above the base. </li></ul>t = 21 – 0.01(300) t = 21 – 3 t = 18 degrees Celsius The ordered pair solution for this equation is (300, 18). 300 is the domain, 18 is the range.
7. Try This One <ul><li>Determine the normal low July temperature at 700m above the base of Mt. Rushmore, using the same equation as before: t = 21 – 0.01n. </li></ul><ul><li>Remember, n = the meters above the base, t = temperature. </li></ul><ul><li>The low temperature in July at Mt. Rushmore is 14 degrees Celsius. </li></ul>
8. Graphing Equations with Two Variables. <ul><li>Equations with two variables can have many solutions..Duh. </li></ul><ul><li>Show ALL of these solutions by graphing . </li></ul><ul><li>Any Equation , that when graphed , forms a line , is a LINEAR EQUATION. </li></ul>
9. Graph: y = -0.5x + 3 <ul><li>Make a table of values to show ordered-pair solutions. </li></ul><ul><li>Graph the ordered pairs. Draw a line through the points. </li></ul>(4, 1) y = -0.5(4) + 3, y = -2 + 3, y = 1 x = 4 (0, 3) y = -0.5(0) + 3, y = 0 + 3, y = 3 x = 0 (-2, 4) y = -0.5(-2) + 3, y = 1 + 3, y = 4 x = -2 Ordered Pairs (x, y) Equation: y = -0.5x + 3 Values for X (that we make up)
10. Graph These For Me In Your Notes, The Connect The Dots <ul><li>y = 2x + 1 </li></ul><ul><li>y = 3x -2 </li></ul>Pick reasonable numbers for x so that you can solve for y and find the ordered pair that allows you to graph the equation. You need three points to graph so that you know you have done it correctly.
11. Linear Equations and Vertical Line Test <ul><li>The graph y = -0.5x + 3 passes the Vertical Line Test. </li></ul><ul><li>Therefore, this equation, the RELATION, is a FUNCTION! </li></ul><ul><li>A Linear Equation is a Function UNLESS its graph is a vertical one. </li></ul><ul><li>…Wait, what does a vertical graph look like? </li></ul>
12. Graph These Equations <ul><li>Y = 2 </li></ul><ul><li>X = 2 </li></ul>For every value of x, y ALWAYS equals 2. No matter what. For every value of y, x ALWAYS equals 2. No matter what. This is the graph for y = 2, it is a horizontal line. Does it pass the vertical line test? This is the graph for x = 2, it is a vertical line. Does it pass the vertical line test? Yes! Y = 2 is a function! No! X = 2 is NOT a function!
13. One More Example <ul><li>Solve for y, then Graph. </li></ul><ul><li>3x + y = -5 </li></ul><ul><li>3x – 3x + y = -5 – 3x (Subtract 3x from both sides) </li></ul><ul><li>y = -3x - 5 (Solved for y) </li></ul><ul><li>Make a Table. </li></ul><ul><li>Graph. </li></ul>(x, y) -3x – 5 X
14. Solve each equation for y, then graph. <ul><li>2x + y = 3 </li></ul><ul><li>y – x = 5 </li></ul><ul><li>-3x + 2y = 6 </li></ul><ul><li>Remember, you come up with your own domain (or x value). Try and make it easy on yourself by picking easy numbers (like: -2, -1, 0, 1, and 2). </li></ul>
15. Assignment #2 <ul><li>Pages 394-395: 1-35 all. </li></ul><ul><li>YOU MUST GRAPH!!! </li></ul><ul><li>Because of the Field Trip, Due Monday . </li></ul>