Problem 49 - Algebra 1

Question

SIMPLIFYING EXPRESSIONS Simplify the expression by combining like terms. $$ 15 x+(-4 x) $$

Step-by-Step Solution

Answer

The simplified expression for the given algebraic expression \(15x + (-4x)\) is \(11x\).

Step 1: Identify Like Terms

In this algebraic expression, the like terms are '15x' and '-4x'. They are termed as like terms because they both contain the same variable, 'x'.

Step 2: Combine Like Terms

To combine these like terms, we carry out the operation (addition or subtraction) on their coefficients. Here, since '-4x' implies 'subtract 4x', we subtract the coefficient of '-4x' from the coefficient of '15x'. This operation is represented as follows: \(15 - 4 = 11\).

Step 3: Write the Simplified Expression

After carrying out this operation, stick the result back with the common variable 'x'. So the simplified expression becomes '11x'. This is the most simplified form of the given algebraic expression, as no further like terms are present to combine.

Key Concepts

Simplifying Expressions

In algebra, simplifying expressions is all about making the expression easier to understand or work with. The primary goal of simplifying an expression is to combine like terms, which reduces the expression to its simplest form. By doing this, you'll make it more manageable when it comes time to solve for unknown variables or integrate it into other equations.

A simplified expression can yield a clearer insight into what the equation represents, allowing you to see relationships between the variables without unnecessary complexity. This process often involves reducing the expression by performing arithmetic operations on coefficients while making sure to preserve the integrity of the expression's values.

When simplifying, always remember to first identify like terms, perform addition or subtraction on their coefficients, and finally, reassemble the expression with the simplified terms. This systematic approach ensures you can simplify any algebraic expression with confidence.

Algebraic Expressions

An algebraic expression is a mathematical phrase comprising numbers, variables, and operation symbols. Unlike simple numerical expressions, algebraic expressions involve variables—a symbol that stands in for an unknown value—that can vary.

Elements of algebraic expressions:

• Constants: Fixed numbers without attached variables, e.g., 7, -3.
• Variables: Letters such as x or y, representing numbers that can change.
• Coefficients: Numbers directly multiplying a variable, denoting how many times the variable is taken, e.g., in 3x, 3 is the coefficient.
• Operators: Symbols denoting operations, like +, -, *, and /.

Understanding these components is crucial, as they allow you to set up equations that model real-world situations.

Algebraic expressions are versatile and can be used to represent almost any mathematical situation, from simple calculations to complex models. Mastering how to interpret and manipulate these expressions is fundamental to excelling in algebra and many branches of mathematics.

Combining Like Terms

Combining like terms is a pivotal technique used to simplify algebraic expressions. It involves merging terms in an expression that have the same variable raised to the same power. This technique is applicable in expressions like '15x + (-4x)'. Here, '15x' and '-4x' are like terms because they both involve the variable 'x'.

Steps to combine like terms:

• Identify the like terms: Look for terms with the same variable and exponent.
• Combine the coefficients: Add or subtract the numerical coefficients of these like terms. For example, from '15x + (-4x)', compute the coefficients as '15 - 4 = 11'.
• Rewrite the expression: Place this new coefficient in front of the shared variable, giving you the simplified term, '11x'.

By following these steps, you ensure that you retain the integrity of the expression while making it more workable.

Combining like terms is essential not only for simplifying expressions but also for solving equations where simplifying the initial complexity makes the next steps more manageable.