Question

State whether or not the given equation is linear. $$x y+y z+x z=1$$

Short answer

The given equation is not linear because it contains products of variables.

Step-by-step solution

Understand the Definition of Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. In other words, linear equations have terms of the form \(a_i x_i\) (where \(a_i\) is a constant and \(x_i\) is a variable) and do not involve any products or powers of variables greater than 1.

Analyze the Given Equation

The given equation is \(x y + y z + x z = 1\). Here, each term involves the product of two variables: \(xy\), \(yz\), and \(xz\). This means that these terms are not of the form \(a_i x_i\) but rather involve products of variables.

Determine Linearity Based on Variable Terms

To identify if the equation is linear, check if all terms are linear terms. Since \(xy\), \(yz\), and \(xz\) are not linear (as they involve the multiplication of variables), the equation \(x y + y z + x z = 1\) does not fit the standard definition of a linear equation.

Key concepts

Algebraic Equation

An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can contain numbers, variables, and operations like addition, subtraction, multiplication, and division.

• Variables in the equation represent unknown values that we aim to solve for.
• The simplest form of an algebraic equation involves one operation. However, they can be much more complex, involving multiple operations and terms.

For example, the equation \(x + 2 = 5\) tells us that the expression \(x + 2\) equals 5, and by solving it, we determine the value of \(x\). When we look at equations, we categorize them into linear and non-linear equations based on their structure.

Variable Multiplication

In algebra, variable multiplication occurs when two or more variables are multiplied together.

• For instance, in the term \(xy\), 'x' and 'y' are multiplied together.
• This is different from a variable multiplied by a constant, such as \(3x\) or \(-5y\), which remains linear.

Variable multiplication plays a crucial role in determining the type of equation. If you find terms like \(xy\), \(yz\), or \(xz\) in an equation, these suggest that it involves the multiplication of variables.This presence indicates that the equation is likely not linear, as linear equations feature terms where each variable is only raised to the first power.

Non-linear Terms

Non-linear terms are parts of an equation that involve anything other than constants, constant-variable products, or variables raised to the power of one.

• Examples include: \(x^2\), \(xy\), and \(x^3\).
• They often suggest a curve or more complex graph when plotted, unlike the straight line characteristic of linear equations.

In the equation \(xy + yz + xz = 1\), the terms \(xy\), \(yz\), and \(xz\) are non-linear because they involve products of variables. Such terms disrupt the linear structure, thus classifying the equation as non-linear. Identifying non-linear terms is essential to understanding the nature of an equation and predicting the kinds of solutions it might have.