Problem 53 - Prealgebra

Question

Koko and John are simplifying the expression \(5 x-4+x+2 .\) Who is correct? Explain your reasoning. Koko \(5 x-4+x+2=\) \(6 x-2\) John \(5 x-4+x+2=\) \(5 x-2\)

Step-by-Step Solution

Answer

Koko is correct: the simplified expression is \(6x - 2\).

Step 1: Combine Like Terms

The expression given is \(5x - 4 + x + 2\). First, we need to combine like terms. There are terms with \(x\), which are \(5x\) and \(x\), and constant terms, which are \(-4\) and \(+2\).

Step 2: Simplify Terms with 'x'

Add the coefficients of the terms with \(x\): \(5x + x = 6x\). So the expression simplifies to \(6x - 4 + 2\).

Step 3: Simplify Constant Terms

Combine the constant terms: \(-4 + 2 = -2\).

Step 4: Write the Simplified Expression

Combine the results of steps 2 and 3 to write the fully simplified expression: \(6x - 2\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental step when simplifying algebraic expressions. Like terms are terms that have the same variables raised to the same powers. For example, in the expression \(5x - 4 + x + 2\), the terms \(5x\) and \(x\) are like terms because they contain the same variable \(x\).
To combine these, you add their coefficients. The coefficients are the numbers in front of the variables which will be discussed later. Here, you add \(5x\) and \(x\) to get \(6x\).

• Remember: only terms with the same variables and powers can be combined.
• Constants, which are terms without variables, are also like terms but in a separate category.

Mastering the ability to identify and combine like terms will make solving algebraic expressions much simpler.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are different from equations because they don't have an equality sign. The expression \(5x - 4 + x + 2\) is an example of an algebraic expression.
Expressions can vary in complexity from simple combinations of numbers and variables to more intricate arrangements.

• It may include addition, subtraction, multiplication, and division.
• Understanding the structure of an expression helps in simplifying it effectively.

Knowing how to break down and simplify algebraic expressions can help you solve more complicated math problems.

Constants

In algebra, constants are terms that do not contain any variables; they are fixed values. For instance, in the expression \(5x - 4 + x + 2\), the constants are \(-4\) and \(+2\).
Constants are crucial because they contribute to the value of an expression once all variables are assigned specific values.

• To simplify expressions involving constants, simply perform the arithmetic operations.
• In the example, we add \(-4\) and \(2\) to get the constant term \(-2\).

Adding constants follows basic arithmetic rules, making it a straightforward part of simplifying expressions.

Coefficients

Coefficients are the numbers that multiply a variable in an algebraic expression. They tell us how many of each variable there are. For example, in the term \(5x\), 5 is the coefficient.
In the expression \(5x - 4 + x + 2\), the coefficients of the terms with \(x\) are \(5\) and an implied \(1\) in \(x\).
Understanding coefficients helps you simplify an expression by combining like terms conveniently. For instance, combining the terms \(5x\) and \(x\) involves adding their coefficients:

• 5 (from \(5x\)) + 1 (from \(x\)) to form \(6x\).

Recognizing and working with coefficients ensures you don't miss important steps in problem-solving.