What you should learn

To determine the range

To graph the solution set for the given domain

NCTM Curriculm Standards 2, 4, 6-10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Equation in two variable

Solution

 

Introduction: Are you as "light" as a feather or as "heavy" as a rock? The density of an object is the measure of its "lightness" or "heaviness". Density is defined as the mass per unit of volume. The table shows the densities of some common substances.

Substance	Density (g/cm)
air aluminum gasoline gold lead mercury silver steel water	0.0013 2.7 0.7 19.3 11.3 13.5 10.5 7.8 1.0

An equation relating density, mass, and volume is m=DV, where m is the mass of the substance, D is the density, and V is the volume. For example, the equation for gold is m=19.3V because the density of gold is 19.3 g/cm. This equation is called an equation in two variables, m and V.

No one knows exactly when gold was first discovered. However, there have been gold artifacts dug up in Ur in Mesoptamia (modern day Iraq) that date as early as 3500 B.C. Suppose we wanted to know the mass of four gold artrifacts with volumes of 10 cm, 50 cm, 100 cm, and 150 cm. We can use a table to find ordered pairs (V, m) that satisfy the equation.

These four ordered pairs are graphed at above. Since each ordered pair satisfies the equation, each ordered pair is a solution of the equation.

Since the solutions of an equation in two variables are ordered pairs, such an equation describes a relation. In an equation involving x and y, the set of x values is the domain of the relation and the set of y values is the range.

Exercise 1: Solve each equation for the given domain values. Graph the solution set.

A. y = 4x if the domain is {-3, -2, 0, 1, 2}

Make a table

The graph of the solution set {(-3, -12), (-2, -8), (0, 0), (1, 4), (2, 8)}

 

 

B. y = x + 6 if the domain is {-4, -3, -1, 2, 4}

Then graph the solution set {(-4,2), (-3, 3), (-1, 5), (2, 8), (4, 10)}

Exercise 2: Solve 8x + 4y = 24 if the domain is {-2, 0, 5, 8}

First solve the equation for y in terms of x

Now do a table and graph like Example 1...

Exercise 3: The equation for the perimeter of a retangle is 2w + 2L = P. Suppose the perimeter of a rectangle is 24 centimeters.

A. Solve the equation for L

B. State the independent and dependent variables and determine the domain and range values for which this equation makes sense.

C. Choose five values for W and find the corresponding values of L.

 

A. First substitute 24 for P in the equation. Then solve for L.

Now solve for L (as done in example 2): L = 12 - W

 

B. Sinc ethe value of L depends on the value of W, L is dependent variable and W is the independent variable. Since distance can only be positive, the values in the domain and range must both be greater than zero.

 

C. Choose any five values for W that work in the equation and seem logical.

 

Activity: EXPLORATION ON GRAPHING CALCULATOR

You can enter selected x values in the TABLE feature of the graphing calculator, and it will calculate the corresponding y values for a given equation.

* Solve the equation for y and enter the equation into the Y+ list.

*Press and use the arrow keys to go to Indpnt: Highlight Ask.

*Press . Scroll up to the top of the X list. Enter the first value of the domain and press . The corresponding Y value appears in the second column. Enter the other values of the domain and record the range values.

 

Your Turn

a. Use a graphing calculator to find the corresponding y values for y = -5x + 12 if x = {-4, -2, 0, 1, 3, 5}. Write the solutions as ordered pairs.

b. Why do you think the Ask feature on the TABLE menu is helpful in finding the ordered pairs that are solutions for equations?

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 13-49 odd, 51-59

Alternative Homework: Enriched: 14-44 even, 45-59

Extra Practice: Students book page 767 Lesson 5-3

Extra Practice Worksheet: Click Here.

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