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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. Operations with algebraic expressions and numerical evaluations are introduced in this chapter. Coefficients are described rather than merely defined. Special binomial products have both literal and symbolic explanations and since they occur so frequently in mathematics, we have been careful to help the student remember them. In each example problem, the student is "talked" through the symbolic form.Objectives of this module: be able to identify the independent and dependent variables of an equation, be able to specify the domain of an equation.

Overview

  • Independent and Dependent Variables
  • The Domain of an Equation

Independent and dependent variables

Independent and dependent variables

In an equation, any variable whose value can be freely assigned is said to be an independent variable. Any variable whose value is determined once the other values have been assigned is said to be a dependent variable. Two examples will help illustrate these concepts.

  1. Consider the equation y = 2 x 7 . If we are free to choose values for x , then x would be considered the independent variable. Since the value of y depends on the value of x , y would be the dependent variable.
  2. Consider the equation m = 4 g k 2 . If we are free to choose values for both g and k , then g and k would be considered independent variables. Since the value of m depends on the values chosen for g and k , m would be the dependent variable.

The domain of an equation

Domain

The process of replacing letters with numbers is called numerical evaluation. The collection of numbers that can replace the independent variable in an equation and yield a meaningful result is called the domain of the equation. The domain of an equation may be the entire collection of real numbers or may be restricted to some subcollection of the real numbers. The restrictions may be due to particular applications of the equation or to problems of computability.

Sample set a

Find the domain of each of the following equations.

y = 2 x , where x is the independent variable.

Any number except 0 can be substituted for x and yield a meaningful result. Hence, the domain is the collection of all real numbers except 0.

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d = 55 t , where t is the independent variable and the equation relates time, t , and distance, d .

It makes little sense to replace t by a negative number, so the domain is the collection of all real numbers greater than or equal to 0.

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k = 2 w w 4 , where the independent variable is w .

The letter w can be replaced by any real number except 4 since that will produce a division by 0. Hence, the domain is the collection of all real numbers except 4.

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a = 5 b 2 + 2 b 6 , where the independent variable is b .

We can replace b by any real number and the expression 5 b 2 + 2 b 6 is computable. Hence, the domain is the collection of all real numbers.

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Practice set a

Find the domain of each of the following equations. Assume that the independent variable is the variable that appears in the expression on the right side of the " = " sign.

y = 5 x + 10

all real numbers

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y = 5 x

all real numbers except 0

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y = 3 + x x

all real numbers except 0

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y = 9 x 6

all real numbers except 6

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m = 1 n + 2

all real numbers except 2

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s = 4 9 t 2 , where this equation relates the distance an object falls, s , to the time, t , it has had to fall.

all real numbers greater than or equal to 0

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g = 4 h 7 21

all real numbers

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Exercises

For the following problems, find the domain of the equations. Assume that the independent variable is the variable that appears in the expression to the right of the equal sign.

y = 4 x + 7

x = all real numbers

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y = x 2 + 2 x 9

x = all real numbers

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y = 11 x

x = all real numbers

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y = 3 x

x = all real numbers except zero

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m = 16 h

h = all real numbers except zero

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t = 5 s 6

s = all real numbers except 6

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Exercises for review

( [link] ) Name the property of real numbers that makes 4 y x 2 = 4 x 2 y a true statement.

commutative property of multiplication

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( [link] ) Simplify x 5 n + 6 x 4 .

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( [link] ) Supply the missing phrase. Absolute value speaks to the question of and not "which way."

"how far"

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( [link] ) Find the product. ( x 8 ) 2 .

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( [link] ) Find the product. ( 4 x + 3 ) ( 4 x 3 ) .

16 x 2 9

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
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2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
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can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
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Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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