Problem 1 - Elementary Algebra

Question

Classify the following equations in terms of their degree. $$ 3 x+6=0 $$

Step-by-Step Solution

Answer

Question: Determine the degree of the equation 3x + 6. Answer: The degree of the given equation is 1, making it a linear equation.

Step 1: Identifying the variable

In the given equation, the variable is x.

Step 2: Determining the power of each term

Now that we have identified the variable, we can look for the power of x in each term of the equation: - In the term "3x", the power of x is 1, as there is no exponent written. - In the term "6", the power of x is 0, as there is no x present in this term.

Step 3: Identifying the highest power of the variable

As we have determined the power of x in each term, we can now identify the highest power. In this case, the highest power of x is 1 (from the term "3x").

Step 4: Classifying the equation based on its degree

The highest power of the variable x in the given equation is 1. Therefore, the degree of the equation is 1. This makes it a linear equation.

Key Concepts

Linear Equations

Understanding linear equations is a fundamental aspect of algebra. A linear equation is a type of algebraic equation that represents a straight line when graphed on a coordinate plane. It has the general form of ax + b = 0, where a and b are constants.

Linear equations have several defining characteristics: firstly, they always have the highest power of the variable, often x, as 1. This means there will be no exponents on the variable or, if written, it is to the first power. Secondly, linear equations result in a straight-line graph because the rate of change of the variable is constant.

In our provided exercise 3x + 6 = 0, we can see it falls perfectly into this definition. The equation simplifies to x = -2 after solving it, showing that for every increase in x, the left-hand side increases by three times the amount, consistent with the definition of a linear equation.

Degree of an Equation

The degree of an equation is an important concept that tells you about the highest power of the variable present in the equation. As a rule of thumb, the degree is associated with the greatest exponent of any term in the algebraic expression.

To find the degree, we look for the term with the highest power of the variable. If an equation has multiple terms, each term's power should be considered separately. The highest one dictates the degree of the entire equation. This aspect can tell a lot about the equation, such as the number of possible solutions and the shape of its graph.

Using our exercise as an example, we have 3x + 6 = 0, where the power of x is 1, making it a first-degree equation. This means it is a linear equation, confirming what we've learned in the previous section.

Algebraic Expressions

Algebraic expressions are the building blocks of algebra and form the basis of forming equations. They are combinations of numbers, variables (like x or y), and arithmetic operations such as addition, subtraction, multiplication, and division. Unlike equations, algebraic expressions are not equal to anything; they are not solved for a specific value but are simplified or manipulated as part of larger problems.

An example of an algebraic expression is 3x + 6 from our exercise. This expression combines the variable x, the constant term 6, and the arithmetic operation of addition. When an algebraic expression is set equal to a value, such as zero, it becomes an equation that we can solve. In this case, our exercise turns the expression into the equation 3x + 6 = 0, which can then be classified and solved accordingly.