Question

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

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

Step-by-step solution

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.

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.

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.

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.

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.

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 \(Ax + By = C\).

Key concepts

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—a 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.