Problem 19 - Prealgebra

Question

Identify the terms, like terms, coefficients, and constants in each expression. \(7 x-3 y+3 z-2\)

Step-by-Step Solution

Answer

Terms: \(7x, -3y, 3z, -2\). No like terms. Coefficients: 7, -3, 3. Constant: -2.

Step 1: Identify the Terms

Terms in an expression are parts that are separated by addition or subtraction signs. In the expression \(7x - 3y + 3z - 2\), the terms are \(7x\), \(-3y\), \(3z\), and \(-2\).

Step 2: Determine Like Terms

Like terms are terms that have the same variable raised to the same power. In this expression, there are no like terms because each term has different variables or no variable at all (as in the constant).

Step 3: Identify the Coefficients

Coefficients are the numbers that multiply the variables in a term. For the terms \(7x\), \(-3y\), and \(3z\), the coefficients are 7, -3, and 3, respectively.

Step 4: Detect the Constant Term

The constant term is a term without a variable, just a number by itself. In this expression, \(-2\) is the constant term.

Key Concepts

Algebraic Expressions

In prealgebra, understanding algebraic expressions is fundamental. An algebraic expression consists of numbers, variables, and operations like addition or subtraction put together. For instance, in the expression \(7x - 3y + 3z - 2\), you can spot numbers, variables (like \(x\), \(y\), and \(z\)), and operators (addition and subtraction).

An algebraic expression can have one or more terms, which are the parts of the expression that are separated by plus or minus signs. These terms are combined using these operations to represent a mathematical concept without a specific equality or value.

Remember, algebraic expressions don't equate to anything like equations, which have an equals sign.

Like Terms

Like terms are components in an algebraic expression that contain the same variables raised to the same power. Recognizing like terms helps simplify expressions and allows for easier computation in future steps.

In the expression \(7x - 3y + 3z - 2\), observe that each term is unique in terms of variable components: \(7x\), \(-3y\), and \(3z\) have different variables, and \(-2\) is constant. Therefore, there are no like terms here. Being familiar with identifying like terms is crucial as it sets the stage for simplifying expressions and solving equations efficiently.

Coefficients

Coefficients are critical numbers in algebraic expressions as they are the numerical part of a term with a variable. They tell you how many times to use the variable in a computation.

In the expression \(7x - 3y + 3z - 2\):

• Term \(7x\) has a coefficient of 7.
• Term \(-3y\) has a coefficient of -3.
• Term \(3z\) has a coefficient of 3.

Coefficients are always multiplied by the variable, and understanding them is essential for many algebra-related tasks, from simplifying expressions to solving equations.

These numbers play a significant role, as they have a direct impact on the value of the expression.

Constants

A constant in an algebraic expression refers to a term without a variable. It remains fixed since it's unaffected by any changes in variables. You're often tasked with identifying and isolating constants when manipulating expressions.

In our expression example \(7x - 3y + 3z - 2\), the constant is \(-2\). Constants stand alone and don't change unless altered by operations combining other constants.

Understanding constants is crucial for distinguishing parts of an expression. This knowledge allows you to clearly see both the fixed values in an expression and those that change with variables.