Problem 3 - Algebra and Trigonometry

Question

\(3-6\) . State whether the equation or system of equations is linear. $$ 6 x-\sqrt{3} y+\frac{1}{2} z=0 $$

Step-by-Step Solution

Answer

Yes, the equation is linear.

Step 1: Identify Linear Equations

Determine whether the given equation is linear by examining its terms. A linear equation only includes variables raised to the first power with no multiplication or division between variables, and no variables inside functions like sine, cosine, square roots, etc.

Step 2: Analyze Each Term

The given equation is \[6x - \sqrt{3}y + \frac{1}{2}z = 0.\]Analyzing the equation term by term:- The variable x is raised to the first power.- The variable y is outside the square root, which is acceptable as long as the square root does not include y.- The variable z is raised to the first power.None of the terms involve any non-linear functions acting on the variables themselves.

Step 3: Conclude Linearity

Since all variables (x, y, z) are to the first power and are not inside any non-linear functions, we can conclude the equation is linear. Non-linear functions of variables like powers greater than one, products of variables, or functions such as trigonometric, exponential, and roots directly applied to variables would make them non-linear. Here, the coefficient \(\sqrt{3}\) is simply a constant multiple of the variable y.

Key Concepts

Variables

In the context of linear equations, a variable is a symbol—typically a letter like \( x \), \( y \), or \( z \)—that represents a number. The value of a variable is not fixed and can change. These are crucial elements in forming linear equations as they help represent unknown values that we aim to find.

• Variables are to the first power in a linear equation.
• They are not multiplied together or divided by other variables.

In linear equations, these variables function as placeholders for numbers, allowing us to form equations to solve for these unknowns.

Non-linear Functions

Linear equations are defined by the absence of non-linear functions acting on their variables. Non-linear functions involve operations that curve the graph of the equation when plotted, such as:

• Raising variables to powers greater than one
• Taking the square root, logarithm, sine, cosine, or any other non-linear mathematical operation directly on variables

In the equation \(6x -\sqrt{3}y + \frac{1}{2}z = 0\), we do not see any such operations applied directly to the variables \(x\), \(y\), or \(z\). This ensures that the equation remains linear.

Constant Multiples

In linear equations, variables can be multiplied by constants. A constant is a fixed number, not a variable. This multiplication is known as a constant multiple. For instance, in the equation \(6x - \sqrt{3}y + \frac{1}{2}z = 0\):

• The constant 6 is multiplied by \(x\).
• The constant \(\sqrt{3}\) is multiplied by \(y\).
• The constant \(\frac{1}{2}\) is multiplied by \(z\).

These constant multiples allow the equation to reflect real-world relationships where one quantity changes at a constant rate with respect to another.

First Power

When we say that variables in a linear equation are raised to the 'first power', we mean that the exponent of the variable is one. This is a critical feature of linear equations. If a variable had an exponent other than one, the equation would be non-linear.
Using our example, the variables \(x\), \(y\), and \(z\) are all raised to the first power in the equation \(6x - \sqrt{3}y + \frac{1}{2}z = 0\). This simplicity ensures that the relationship represented by the equation is straightforward, leading to a straight line when graphed, which is the hallmark of linearity.