© William James Calhoun, : Equations as Relations OBJECTIVES: You must be able to determine the range for a given domain and graph the solution set for the given domain. We have discussed and solved equations that involved one variable. These equations were solved by adding, subtracting, multiplying, and dividing both sides of the equation by numbers. The goal was to get the variable alone on one side and a number on the other side. That number was the value the variable was holding the place for. That number was the only value the variable could be. It was a definite thing - ONE SOLUTION.

© William James Calhoun, : Equations as Relations An equation in TWO variables essentially sets one variable equal to another. Each variable could represent any number. As one of the variables changes its value, the value of the other variable will change. There is no one, simple answer for these equations, since the variables can change infinitely. Before, on the one-variable equations, the solution to the equation was easy to define. It was the number yielded at the end of the process of isolating the variable. Now we need a new definition for a solution with a two-variable equation.

© William James Calhoun, : Equations as Relations If a true statement results when the numbers in an ordered pair are substituted into an equation in two variables, then the ordered pair is a solution of the equation DEFINITION OF THE SOLUTION OF AN EQUATION IN TWO VARIABLES The one-variable equations looked like this: 3x + 4 = 9 Two-variable equations look the same, only with two variables: y = x + 6 There is still an “=“ sign that must maintain balance between the sides. You can still add, subtract, multiply, and divide both sides by any number. You can still move things around so that you end up with an isolated variable on one side. When you do isolate a variable, there is another one on the other side.

© William James Calhoun, : Equations as Relations Remember that the domain is the independent variable which is generally represented by “x”. EXAMPLE 1: Solve y = 4x if the domain is {-3, -2, 0, 1, 2}. Then graph the solution set. First thing to do with this problem is make a table of the values. They have already told us the values for x (domain). Next we have to determine the range. Each value of the range, y, is four times the x-value. Then we write the solutions as ordered pairs.Solution set: {(-3, -12), (-2, -8), (0, 0), (1, 4), (2, 8)} Now plot the points. Since they gave us a specific domain, we do not connect the dots. Domain x x 4(-3) 4(-2) 4(0) 4(1) 4(2) Range y Ordered Pair (x, y) (-3, -12) (-2, -8) (0, 0) (1, 4) (2, 8) Also, the range is the dependent variable which is represented by “y”.

© William James Calhoun, : Equations as Relations EXAMPLE 2: Solve y = x + 6 if the domain is {-4, -3, -1, 2, 4}. Then graph the solution set. Again, make a table with the given x-values, determine the y-values and write the solution as ordered pairs. Then plot the points. The solution set is: {(-4, 2), (-3, 3), (-1, 5), (2, 8), (4, 10)}. Sometimes, you will need to solve for y first, then set up a table of values. The next example is like this.

© William James Calhoun, : Equations as Relations EXAMPLE 3: Solve 8x + 4y = 24 if the domain is {-2, 0, 5, 8}. The solution to a two-variable equation is a set of ordered pairs. We already have the first number in each ordered pair, the domain (x-value.) Now we need the second number in each ordered pair, the range (y-value.) Since we a looking to find y, solve the equation for y. This means get y on a side by itself. 8x + 4y = 24 4y = x -8x 4 y = 6 - 2x Now make a table. x x 6 - 2(-2) 6 - 2(0) 6 - 2(5) 6 - 2(8) y (x, y) (-2, 10) (0, 6) (5, -4) (8, -10) This is your solution. {(-2, 10), (0, 6), (5, -4), (8, -10)}

© William James Calhoun, 2001 The variable used in two-variable equations will not always be x and y. 5-3: Equations as Relations Sometimes it is obvious which variable is the domain and which is the range. Other times, how the equations are written will determine which variable stands for the domain and which variable stands for the range. On any test I give you, x will be the domain and y will be the range. Some problems in the book expect you to assume the domain is the letter closest to “A” in the alphabet. The letter furthest from “A” in the alphabet is the range. Unfortunately, this little rule does not always work.

© William James Calhoun, : Equations as Relations HOMEWORK Page 275 # odd