Problem 14 - Precalculus Mathematics for Calculus

Question

Find the sum, difference, or product. $$(5-3 x)+(2 x-8)$$

Step-by-Step Solution

Answer

The sum is \(-x - 3\).

Step 1: Identify Like Terms

The expression given is \((5 - 3x) + (2x - 8)\). We need to find the sum of two binomials by first identifying and grouping like terms. Like terms in this case are constants and terms with the same variable powers. Hence, group constants together:\[ 5 + (-8) \] and variable terms together: \[ -3x + 2x \].

Step 2: Sum the Constants

Add the constant terms together. Here, \(5 + (-8)\) is equivalent to \(5 - 8\), which simplifies to \(-3\).

Step 3: Combine the Variable Terms

Now, add the coefficients of the \(x\) terms. The expression \(-3x + 2x\) simplifies to \((-3+2)x\), which equals \(-1x\) or simply \(-x\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to form the simplified expression: \(-x - 3\).

Key Concepts

Like Terms

When working with algebraic expressions, identifying like terms is key to simplifying them. Like terms are terms that have the exact same variable part, including the power to which the variable is raised. For example, in the expression \(4x + 5x - 3\), both \(4x\) and \(5x\) are like terms because they both have the variable \(x\) raised to the same power (which is 1 here).
On the other hand, \(4x\) and \(3x^2\) are not like terms since the powers of \(x\) differ (\(x\) versus \(x^2\)). Recognizing these terms allows us to combine them, simplifying the expression.

• Identify terms with the same variable.
• Check if the variables are raised to the same power.
• Group these terms together for simplification.

Binomials

A binomial is an algebraic expression consisting of exactly two terms separated by a plus or minus sign. These terms can be constants, variables, or a mix of both. For instance, \( (5-3x) \ and \ (2x-8) \) are examples of binomials.
Binomials are an elementary building block in algebra, and understanding them is crucial when it comes to operations like addition, subtraction, and multiplication of algebraic expressions.

• Each binomial has two distinct terms.
• Can involve constants, variables, or both.
• Usually separated by a plus or minus sign.

Simplifying Expressions

Simplifying expressions involves combining like terms to make the expression as concise and readable as possible. This process often includes adding, subtracting, or multiplying terms and then reducing them to their simplest form. From our original expression \( (5 - 3x) + (2x - 8) \), simplifying required us to:

• Identify and group like terms.
• Add or subtract the constants and coefficients of like terms.
• Write down the simplified expression correctly.

By doing this, the simplified expression becomes \(-x - 3\), showing a cleaner version of what might initially seem complex.
Simplifying makes calculations easier and reduces the risk of mistakes in further algebraic manipulations.