4.8: Terminology Associated with Equations

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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.

Consider the equation \(y=2x−7\). 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.
Consider the equation \(m=−4gk^2\). 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.

Example \(\PageIndex{1}\)

\(y = \dfrac{2}{x}\), 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.

Example \(\PageIndex{2}\)

\(d = 55t\), 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.

Example \(\PageIndex{3}\)

\(k = \dfrac{2w}{w-4}\), 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.

Example \(\PageIndex{4}\)

\(a = 5b^2 + 2b - 6\), where the independent variable is \(b\).

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

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.

Practice Problem \(\PageIndex{1}\)

\(y = 5x + 10\)

Answer: All real numbers

Practice Problem \(\PageIndex{2}\)

\(y = \dfrac{5}{x}\)

Answer: all real numbers except 0

Practice Problem \(\PageIndex{3}\)

\(y = \dfrac{3+x}{x}\)

Answer: All real numbers except 0

Practice Problem \(\PageIndex{4}\)

\(y = \dfrac{9}{x-6}\)

Answer: All real numbers except 6

Practice Problem \(\PageIndex{5}\)

\(m = \dfrac{1}{n+2}\)

Answer: All real numbers except \(-2\).

Practice Problem \(\PageIndex{6}\)

\(s = \dfrac{4}{9}t^2\), where this equation relates the distance an object falls, \(s\), to the time, \(t\), it has had to fall.

Answer: All real numbers greater than or equal to 0

Practice Problem \(\PageIndex{7}\)

\(g = \dfrac{4h-7}{21}\)

Answer: All real numbers

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.

Exercise \(\PageIndex{1}\)

\(y = 4x + 7\)

Answer: \(x\) = all real numbers

Exercise \(\PageIndex{2}\)

\(y = 3x - 5\)

Exercise \(\PageIndex{3}\)

\(y = x^2 + 2x - 9\)

Answer: \(x\) = all real numbers

Exercise \(\PageIndex{4}\)

\(y = 8x^3 - 6\)

Exercise \(\PageIndex{5}\)

\(y = 11x\)

Answer: \(x\) = all real numbers

Exercise \(\PageIndex{6}\)

\(s = 7t\)

Exercise \(\PageIndex{7}\)

\(y = \dfrac{3}{x}\)

Answer: \(x\) = all real numbres except zero

Exercise \(\PageIndex{8}\)

\(y = \dfrac{2}{x}\)

Exercise \(\PageIndex{9}\)

\(m = \dfrac{-16}{h}\)

Answer: \(h\) = all real numbers except zero

Exercise \(\PageIndex{10}\)

\(k = \dfrac{4t^2}{t-1}\)

Exercise \(\PageIndex{11}\)

\(t = \dfrac{5}{s-6}\)

Answer: \(s\) = all real numbers except 6

Exercise \(\PageIndex{12}\)

\(y = \dfrac{12}{x+7}\)

Exercises for Review

Exercise \(\PageIndex{13}\)

Name the property of real numbers that makes \(4yx^2 = 4x^2y\) a true statement.

Answer: commutative property of multiplication

Exercise \(\PageIndex{14}\)

Simplify \(\dfrac{x^{5n+6}}{x^4}\)

Exercise \(\PageIndex{15}\)

Supply the missing phrase. Absolute value speaks to the question of and not "which way."

Answer: "how far"

Exercise \(\PageIndex{16}\)

Find the product. \((x-8)^2\).

Exercise \(\PageIndex{17}\)

Find the product. \((4x+3)(4x-3)\).

Answer: \(16x^2 - 9\)