Question

State whether or not the given equation is linear. $$-3 x+9=3 y-5 z+x-7$$

Short answer

The equation is linear, as all terms are of degree 1.

Step-by-step solution

Identify all variables in the equation

The given equation is \(-3x + 9 = 3y - 5z + x - 7\). Identify the variables present, which are \(x\), \(y\), and \(z\).

Rearrange the equation

Rearrange the equation to isolate similar terms on both sides. \(-3x + 9 - x + 7 = 3y - 5z\)which simplifies to\(-4x + 16 = 3y - 5z\).

Check the degree of each variable

Linear equations have each term of the form \(a_1x_1 + a_2x_2 + ... + a_nx_n = b\), where all exponents are 1. In this equation, all variables \(x\), \(y\), and \(z\) have a degree of 1.

Determine if the equation is linear

Since all the variables in the rearranged equation \(-4x + 16 = 3y - 5z\) are linear with no products or powers of variables (i.e., highest degree is 1), the given equation fits the definition of a linear equation.

Key concepts

Variables Identification

In any mathematical equation, recognizing the variables is the foundational step. In the original equation \(-3x + 9 = 3y - 5z + x - 7\), identifying these variables helps to understand the structure of the equation. Variables are symbols that represent numbers or values that can change. In this equation, we have three different variables: \(x\), \(y\), and \(z\). Their presence shows us that the problem might involve more than one dimension or component.

This step sets the stage for further simplification. Ensure that you've correctly identified each variable, as confusing one variable for another might lead to errors in solving or understanding the equation.

Equation Rearrangement

Rearranging an equation is a crucial technique that allows for easier analysis and solution-finding. In our case, the original equation is \(-3x + 9 = 3y - 5z + x - 7\). The objective of rearrangement is to gather like terms, which are terms that contain the same variable raised to the same power.

In our example, we move terms with the same variable on one side:

• Combine \(-3x\) and \(+x\) to get \(-4x\).
• Add constant numbers \(9\) and \(-7\) together to balance the equation.

This leads to the rearranged equation: \(-4x + 16 = 3y - 5z\). Such rearrangement prepares the equation for the next steps and makes it easier to check for specific properties such as linearity.

Degree of Variables

The degree of a variable in an equation is the highest power to which the variable is raised. This concept is quite pivotal when classifying equations. In linear equations, each variable should have a degree of 1. Let's consider our rearranged equation: \(-4x + 16 = 3y - 5z\).

In this equation, each term involving a variable (\(-4x\), \(3y\), and \(-5z\)) has a degree of 1, since they are all raised to the power of one. This characteristic confirms that the equation is linear. Recognizing the degree of variables helps in verifying the type of equation you are dealing with and directs the method of solving it.

Linear Equation Properties

Linear equations have distinct properties that differentiate them from other types of equations. In a linear equation like \(-4x + 16 = 3y - 5z\), the following characteristics apply:

• Each term is either a constant or a product of a constant and a single variable.
• No variable is raised to a power higher than one, ensuring each variable is linear.
• There are no products of variables with each other, which means terms like \(xy\) or \(xz\) are absent.

Because our equation meets all these conditions, it can be classified as a linear equation. Understanding these properties helps when predicting the behavior of the equation and provides clarity on the kind of solutions expected.