Problem 48 - Fundamentals of Mathematics

Question

For problems \(47-56\), simplify each expression by combining like terms. $$ 7 x+3 x-14 x $$

Step-by-Step Solution

Answer

The simplified expression is \(-4x\).

Step 1: Identify Like Terms

The expression given is \(7x + 3x - 14x\). We start by identifying like terms. Like terms are the terms that have the same variable raised to the same power. Here, all the terms \(7x\), \(3x\), and \(-14x\) are like terms because they all involve the variable \(x\) raised to the first power.

Step 2: Combine Like Terms

To simplify the expression, add or subtract the coefficients of the like terms. This gives us \((7 + 3 - 14)x\). Compute the sum inside the parentheses: \(7 + 3 = 10\) and then \(10 - 14 = -4\). Therefore, the simplified form of the expression is \(-4x\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental concept in algebra that helps in simplifying expressions. In any algebraic expression, like terms refer to terms that have the exact same variable raised to the same power. For example, in the expression \(7x + 3x - 14x\), all three terms are like terms because they share the variable \(x\) with no exponent other than 1. Like terms can be combined by performing basic arithmetic operations such as addition or subtraction on their coefficients.

• Step 1: Identify the like terms in the expression.
• Step 2: Add or subtract the coefficients of these terms while keeping the common variable.

For the expression given, \(7x + 3x - 14x\), we add the coefficients \(7\) and \(3\), and then subtract \(14\), giving us \((7 + 3 - 14)x = -4x\). This process not only simplifies the expression but also makes it easier to understand and manipulate in further calculations.

Simplifying Expressions

Simplifying expressions is an essential part of algebra that involves rewriting expressions in a cleaner and more understandable form. This usually involves combining like terms, but it can also include performing operations such as factoring, distributing, or applying mathematical properties to maintain equality.
When simplifying the expression \(7x + 3x - 14x\), we focus first on combining like terms as these share a common variable, \(x\). After combining the coefficients of like terms, the expression reduces to \(-4x\). The goal of simplifying expressions is to make them easier to use in further computations by eliminating unnecessary complexity.
Consider these tips:

• Always combine like terms first to reduce complexity.
• Simplify every coefficient and constant as much as possible.
• Double-check your operations to ensure accuracy.

By following these steps, you create a streamlined version of the algebraic expression that is ready for further analysis or use in solving equations.

Coefficients in Algebra

Coefficients play a crucial role in algebra as they are the numerical factors that multiply the variables in an expression. Understanding coefficients is essential when working on combining like terms and simplifying expressions. In the expression \(7x + 3x - 14x\), the coefficients are \(7\), \(3\), and \(-14\) respectively.

• The coefficient tells us how many times to multiply the variable it accompanies.
• Like terms have the same variable, but possibly different coefficients.
• To combine like terms, focus on adding or subtracting these coefficients while keeping the variable unchanged.

For example, in \(7x + 3x - 14x\), we add the first two coefficients to get \(10\), then subtract \(14\) to get \(-4\). Hence, the simplified expression becomes \(-4x\).
In summary, coefficients help determine the calculation of terms in expressions. By mastering the manipulation of coefficients, you will enhance your ability to simplify expressions significantly.