Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

Simplify each algebraic expression. $$7 x+8+2 x-3$$

Step-by-Step Solution

Answer

The simplified algebraic expression is \(9x + 5\).

Step 1: Combining Like Terms

In the algebraic expression \(7 x+8+2 x-3\), there are two terms with the variable 'x' and two constant terms. These 'like terms' can be combined separately. Combine \(7x\) and \(2x\) to get \(9x\) then combine \(8\) and \(-3\) to get \(5\).

Step 2: Write the Simplified Expression

The combined 'like terms' give us the simplified algebraic expression, which is \(9x + 5\).

Key Concepts

like terms

In algebra, the concept of 'like terms' is foundational for simplifying expressions. 'Like terms' are terms that have the same variable raised to the same power. For instance, in the expression given, \(7x\) and \(2x\) are 'like terms' because they both contain the variable \(x\) raised to the same power (the first power). Identifying like terms allows us to combine them into a single term. Constant numerical terms, such as \(8\) and \(-3\), are also 'like terms' since they do not involve variables.
Recognizing like terms is essential before performing any algebraic operations, as it sets the stage for simplification. It means scanning each part of an expression and determining which terms share common factors or variables. Once you can spot these elements, the process of algebraic manipulation becomes straightforward and efficient.

combining terms

Combining terms is the next step after identifying like terms. The process involves adding or subtracting coefficients of the like terms. In our exercise, once \(7x\) and \(2x\) are identified as like terms, we combine them by adding their coefficients: \(7 + 2\). This gives us \(9x\). Similarly, the constants \(8\) and \(-3\) are combined to form \(5\) through simple arithmetic.
Here are some tips to make combining terms seamless:

• Always group terms with the same variables and exponents first.
• Ensure to carefully add or subtract their numerical coefficients.
• Reorder the terms if it helps visually to pair them off correctly.

After combining, the expression is more compact and easier to understand or use in further calculations.

simplified expression

Once terms have been combined, the final product is the simplified expression. In our example, after combining, we obtain \(9x + 5\). A simplified expression is a cleaner, more efficient version of the original, devoid of unnecessary terms. It maintains the integrity of the mathematical relationship but presents it in its simplest form.
This simplification does not change the value of the expression; it merely makes it easier to interpret and work with. As students practice simplifying expressions, they develop a keener sense of how variables interact and how to manage numbers efficiently. The goal is always to arrive at the most easily understandable form possible, allowing for swift calculations or easier application in broader algebraic problems. Simplification is a skill that, once mastered, underpins more advanced mathematical concepts.