Problem 43 - Elementary Algebra

Question

For the following problems, classify each of the equations by degree. If the term linear, quadratic, or cubic applies, use it. $$ y-x-z+4 w=21 $$

Step-by-Step Solution

Answer

Answer: The degree of the given equation is \(1\). Based on its degree, it can be classified as linear.

Step 1: Observe the given equation

We are given an equation with four variables: $$ y-x-z+4w=21 $$

Step 2: Identify the exponents of each variable

Now, let's look at the exponents of the variables in the given equation: - The exponent of \(y\) is \(1\) (since there are no visible exponents, it's implicitly \(1\)) - The exponent of \(x\) is \(1\) (same as above) - The exponent of \(z\) is \(1\) (same as above) - The exponent of \(w\) is \(1\) (same as above)

Step 3: Determine the degree of the equation

According to the exponents, the highest exponent in the equation is \(1\). Thus, the degree of the equation is \(1\).

Step 4: Classify the equation based on the degree

Since the degree of the equation is \(1\), the given equation can be classified as linear.

Key Concepts

Linear Equations

Linear equations are mathematical statements involving algebraic expressions set equal to a specific value. A key feature of linear equations is that all of their variables appear only to the first power. This means that each variable is not raised to any power greater than one.
For example, in the equation \( y - x - z + 4w = 21 \), each term with a variable is a linear term because their exponents are all \( 1 \).

• Linear equations often represent straight lines when graphed on a coordinate plane.
• They can include multiple variables, but each variable's highest exponent must be \( 1 \).
• The equation may contain constants, like \( 21 \) in this case, which shifts the line but does not affect its linearity.

Linear equations are foundational in algebra, forming the basis for more complex mathematical concepts and operations.

Degree of Equation

The degree of an equation is determined by the highest power of any variable present within it. Understanding the degree of an equation is essential as it informs us of the nature and behavior of the equation itself. - For the given equation \( y - x - z + 4w = 21 \), note that all variables are to the first power. - This indicates that the equation's degree is \( 1 \).

• A degree of \( 1 \) indicates a linear equation.
• Quadratic equations have a degree of \( 2 \), characterized by terms like \( x^2 \).
• Cubic equations have a degree of \( 3 \), seen in terms such as \( x^3 \).

Recognizing the degree allows for classification into linear, quadratic, cubic, or higher polynomial types, aiding in understanding how solutions and graphs behave.

Algebraic Variables

Algebraic variables are symbols that represent unknown or arbitrary numbers in a mathematical expression or equation. In our example equation \( y - x - z + 4w = 21 \), the symbols \( y \), \( x \), \( z \), and \( w \) are the algebraic variables that hold unknown values.
Variables are crucial as they allow for general formulations of mathematical concepts and operations.

• They provide a way to express relationships and dependencies between various quantities.
• The coefficients, such as \( -1 \) for \( x \) or \( 4 \) for \( w \), indicate the weight or impact each variable has within the equation.
• Variables enable equations to be flexible and applicable to solve numerous practical situations.

Understanding the role of algebraic variables is essential in forming equations and interpreting mathematical models used in various fields.