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Chapter 13: Problem 4

Which of the following implicitly defines a linear function? A. \(x^2+y^2=25\) B. \(3 x-5 y=9\) C. \(y=\frac{1}{3 x}-2\) D. \(x y=8\)

Short Answer

Expert verified
Option B. \(3x - 5y = 9\)

Step by step solution

01

Understand Linear Functions

A linear function is an equation that can be written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants, and \(x\) and \(y\) are variables. The graph of a linear function is a straight line.
02

Evaluate Option A

Check if the equation \(x^2 + y^2 = 25\) is linear. The presence of \(x^2\) and \(y^2\) indicates that this is a nonlinear equation. Therefore, it does not define a linear function.
03

Evaluate Option B

Check if the equation \(3x - 5y = 9\) is linear. This equation is in the form \(Ax + By = C\) with \(A = 3\), \(B = -5\), and \(C = 9\). Thus, it defines a linear function.
04

Evaluate Option C

Check if the equation \(y = \frac{1}{3x} - 2\) is linear. The term \(\frac{1}{3x}\) (which is the same as \(x^{-1}\)) implies a reciprocal relationship, making this a nonlinear function. Thus, it does not define a linear function.
05

Evaluate Option D

Check if the equation \(xy = 8\) is linear. This equation represents a product of variables, which is not in the form \(Ax + By = C\). Hence, it is also nonlinear and does not define a linear function.
06

Conclusion

Among the given options, only \(3x - 5y = 9\) meets the criteria for defining a linear function, as it can be written in the form \(Ax + By = C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations are foundational elements in mathematics and have a straightforward form:
  • They can always be expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
  • In this equation, \(x\) and \(y\) are variables.
  • The graph representing a linear equation is a straight line.
Understanding the characteristics of linear equations makes it easier to identify and solve them. Keep in mind that linear equations will always render straight-line graphs, helping you quickly distinguish them from nonlinear equations.
Nonlinear Equations
Nonlinear equations differ significantly from linear equations as they don't form straight lines on a graph. Here are some key points to grasp about nonlinear equations:
  • They can involve exponents, roots, or products of variables.
  • Their general forms do not adhere to the structure \(Ax + By = C\).
  • The resulting graphs can be curves, parabolas, circles, or other shapes.
For instance, consider the equation \(x^2 + y^2 = 25\), which represents a circle鈥攁 typical trait of nonlinear equations. Identifying such characteristics helps determine whether an equation is nonlinear, thus aiding in avoiding common mistakes.
Graph of a Linear Function
The graph of a linear function provides insight into the nature of the equation it represents. Key facts to note:
  • A linear function graph will always be a straight line.
  • The slope of this line tells us how steep the line is.
  • The y-intercept shows where the line crosses the y-axis.
For example, the linear equation \(3x - 5y = 9\) can be rearranged to the slope-intercept form using \(y = mx + b\): \[y = \frac{3}{5}x - \frac{9}{5}\]. Here, the slope \(m = \frac{3}{5}\) and y-intercept \(b = -\frac{9}{5}\) guide us in plotting the graph.
Equation Evaluation
Evaluating equations is key to distinguishing their types. There are some steps to make this process easy:
  • Check for the standard linear form \(Ax + By = C\).
  • Look for terms involving squares, cubes, or products of variables to identify nonlinear functions.
  • Remember that terms involving reciprocals (like \(\frac{1}{3x}\)) denote nonlinear equations.
Reviewing the given options, only \(3x - 5y = 9\) is evaluated to be linear since it satisfies \(Ax + By = C\). Applying these evaluation steps ensures accurate identification and understanding of equations.

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Most popular questions from this chapter

Find \(f(-3)\) for \(f(x)=-2 x^2-7 x+9\) A. -30 B. -18 C. 12 D. 48

A set D has 5 elements, and a set \(R\) has 3 elements. How many functions can be defined with domain \(D\) and range \(R\) ? A. 15 B. 125 C. 150 D. 243

Line A passes through the point \((0,17)\) on the \(y\)-axis and continues to the right, dropping 3 units for every unit to the right. Line \(B\) is given by \(5 x-3 y=30\). Which line has the larger \(x\)-intercept? $$ \begin{array}{|l|l|} \hline \text { Select... } & \nabla \\ \hline \text { Line A } & \\ \hline \text { Line B } & \\ \hline \end{array} $$

The equation \(f(a)=b\) implies that what point is on the graph of \(f(x)\) ? A. \((a, b)\) B. \((b, a)\) C. \((-a, b)\) D. \((-b, a)\)

The function \(I(n)=28,000+4,000 n\) represents the average annual income in dollars for a person with \(n\) years of college education. What is the best interpretation for the equation \(I(4)=44,000\) ? A. A person with 4 years of college should request an annual average salary of \(\$ 44,000\) when interviewing. B. A person with 4 years of college will earn \(\$ 44,000\) more each year on average than if they didn't attend college. C. A person with 4 years of college will earn \(\$ 44,000\) annually, on average. D. A person with 4 years of college should look for a position that starts at \(\$ 44,000\) annually.
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