Question

State whether or not the given equation is linear. $$x+y+z=10$$

Short answer

The equation \(x + y + z = 10\) is linear.

Step-by-step solution

Understand the definition of a linear equation

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. It can be written in the form \(a_1x_1 + a_2x_2 + ... + a_nx_n = b\), where \(a_1, a_2, ..., a_n\) and \(b\) are constants.

Analyze the given equation

The given equation is \(x + y + z = 10\). Here, \(x, y, \) and \(z\) are variables, and the constant term is 10. Each variable appears with a linear coefficient of 1, which is considered a constant.

Compare with the linear equation standard form

Compare the given equation \(x + y + z = 10\) with the standard form \(a_1x_1 + a_2x_2 + ... + a_nx_n = b\). In this equation, \(a_1 = 1\), \(a_2 = 1\), \(a_3 = 1\), and \(b = 10\). All terms match the form of a linear equation, as there are no products of variables or variables raised to a power other than one.

Conclude whether the equation is linear

Since \(x + y + z = 10\) matches the conditions of a linear equation—each term is linear and no exponents are higher than 1—it is a linear equation.

Key concepts

Algebraic Equations

An algebraic equation is a mathematical statement that shows the equality between two expressions. These expressions consist of numbers, variables, and operations. In algebraic equations, the balance between both sides of the equation is key. When you manipulate one side, you must do the same to the other.

Algebraic equations can come in different forms, such as:

• Polynomial equations
• Exponential equations
• Rational equations

Polynomial equations, like linear equations, involve terms with variables raised to a power. The power on the variable in a linear equation, however, is always one because it is the simplest form of an algebraic equation. To solve an algebraic equation means to find the value or values for the variables involved that make the equation true.

Variables and Constants

In algebra, equations are composed of variables and constants. A variable is a symbol used to represent an unknown number, often depicted as letters like \(x\), \(y\), or \(z\). Variables are crucial because they allow equations to have flexibility and encompass many possible values.

A constant, on the other hand, is a fixed number that does not change. In the equation \(x + y + z = 10\), the number \(10\) is a constant. Constants provide a reference point or set a condition for the equation. Think of variables as the wild cards and constants as the rules they must follow.

Understanding the roles of variables and constants is essential for manipulating and solving algebraic equations. When solving equations, we often aim to isolate the variable to one side and have the constants on the other.

Standard Form of Linear Equations

The standard form of a linear equation is a widely used expression, \(a_1x_1 + a_2x_2 + ... + a_nx_n = b\), where all the constants \(a_1, a_2, \ldots, a_n,\) and \(b\) must be identifiable.

This form ensures that each variable is multiplied by a constant coefficient and then added together. It concludes with an equality to a final constant (\(b\)), which is essential for determining the validity and linearity of the equation.

• The coefficients \(a_1, a_2, ..., a_n\) must not be zero.
• Each variable, represented as \(x_1, x_2, ..., x_n\), should be raised to the first power.

In real-world applications, this standard form makes it easier to solve systems of equations, graph equations, and handle algebraic operations involving multiple equations. Identifying a linear equation's standard form helps verify the nature of the equation, as seen in the example \(x + y + z = 10\).