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Chapter 3: Problem 11

Determine whether each equation is linear. $$ x y=7 $$

Short Answer

Expert verified
The equation xy = 7 is not linear.

Step by step solution

01

- Define a Linear Equation

A linear equation in two variables is an equation of the form ewline en: ax + by = c, where a, b, and c are constants.
02

- Analyze the given equation

Consider the given equation: xy = 7. In this form, the equation is the product of the variables x and y.
03

- Check for linearity

For an equation to be linear (in standard form), each variable should be of the power of 1 and only involved in an addition or subtraction operation鈥攏o multiplication of variables. Since x and y are multiplied together in xy = 7, the equation is not linear.

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Key Concepts

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

analyzing equations
Analyzing linear equations is essential to understand their behavior and solutions. When you analyze a given equation, start by identifying its structure and the operations it involves. For example, linear equations typically adhere to a specific form. Identifying whether the variables are only added, subtracted, or multiplied by constants can help classify the equation correctly. If variables are involved in more complex operations, like multiplication by each other, exponents, or other non-linear terms, the equation is likely not linear. Knowing these distinctions allows you to solve the given equations more effectively and grasp their underlying patterns better.
variables
Variables are symbols, usually letters, representing unknown values in equations. In the context of linear equations, variables typically stand for values that will make the equation true when substituted with real numbers. For instance, in the equation ax + by = c, 'x' and 'y' are variables. Their values can vary, hence the term 'variable'. Recognizing how these variables interact within an equation is crucial. In a linear equation, each variable should be to the power of one. This means they should not be squared, cubed, or involved in any non-linear operations like multiplication of one variable by another, such as xy = 7, which makes the equation non-linear.
standard form
The standard form of a linear equation in two variables is written as ax + by = c. Here, a, b, and c are constants, and x and y are variables. This form is useful because it provides a consistent way to look at and solve linear equations. To check if a given equation is in standard form, analyze its structure to see if it can be manipulated into this format. For instance, equations like 2x + 3y = 12 or -x + y = 5 are in standard form, and you can easily see the relationship between x and y. Remember, if the equation involves expressions like x^2, y^2, or xy, it is not in standard form and hence not a linear equation.

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Most popular questions from this chapter

To prepare for Section \(3.7,\) review solving a formula for \(a\) variable and subtracting real numbers (Sections 1.6 and 2.3 ). Simplify. $$-4-5$$

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