Problem 65 - Elementary Algebra Within Reach

Question

In Exercises \(63-68\), simplify the expression. $$ 6 x+9 x $$

Step-by-Step Solution

Answer

The simplified expression is \(15x\).

Step 1: Identify Like Terms

In the given expression \(6x + 9x\), both terms are like terms because they both have the variable \(x\).

Step 2: Add the Coefficients

To simplify the expression, add the coefficients of the like terms together. The coefficient of \(6x\) is 6 and the coefficient of \(9x\) is 9. Add the coefficients 6 + 9 to get 15.

Step 3: Write the Simplified Expression

Place the sum of the coefficients, 15, in front of the variable \(x\) to get the simplified expression: \(15x\).

Key Concepts

Like Terms

When working with algebraic expressions, like terms are crucial to simplifying an expression easily. Like terms have the same variable part; in essence, they share identical variables raised to the same power. In the expression \(6x + 9x\), both terms are like terms due to the presence of the variable \(x\) without any exponents or different variables involved. Recognizing like terms allows mathematicians and students to combine them, making expressions more concise and manageable.

• Like terms have the same variable letter, such as \(x\) in both terms.
• They also have the same exponent, even if it's not explicitly written (exponent of 1 in this case).
• Combining like terms simplifies calculations in algebraic manipulations.

Coefficients

In any algebraic expression, a coefficient is the numerical factor that multiplies a variable. Coefficients provide the scale or multiple for the variable part. In the expression \(6x + 9x\), the numbers 6 and 9 are the coefficients. These coefficients are crucial when combining like terms, as they determine how many times or to what extent the variable is included in the sum. When simplifying expressions, follow these steps:

• Identify the coefficients associated with each like term.
• Add or subtract these coefficients as needed, based on the operation given in the expression.
• Recombine the number with the variable to form a simplified term.

In our example, the coefficients 6 and 9 are added together to become 15, which then scales the variable \(x\) in the simplified expression \(15x\).

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They form the cornerstone of algebra, allowing us to generalize mathematical problems and solutions. Such expressions can range from simple binomials like \(6x + 9x\) to more complex polynomials.Key components of algebraic expressions include:

• Numbers and constants, which provide fixed values.
• Variables, which stand for unknown or changing quantities.
• Operations such as addition, subtraction, multiplication, or division.

To simplify an expression:

• Identify any like terms to be combined.
• Compute the operations specified, often involving coefficients.
• Reduce the expression to its simplest form for easy usage in calculations.

Simplifying expressions like \(6x + 9x\) involves combining like terms and correctly applying arithmetic rules to arrive at a clearer and more condensed form.