Problem 5 - Prealgebra

Question

Simplify expression. \(x+9 x+3\)

Step-by-Step Solution

Answer

The simplified expression is \(10x + 3\).

Step 1: Combine Like Terms

The expression given is \(x + 9x + 3\). First, identify and combine the like terms. The like terms here are \(x\) and \(9x\), which are both terms containing the variable \(x\). Add them together: \(x + 9x = 10x\).

Step 2: Rewrite the Simplified Expression

Now that the like terms \(x\) and \(9x\) have been combined to form \(10x\), rewrite the expression. The new expression is \(10x + 3\). This expression does not contain any further like terms to simplify.

Key Concepts

Combining Like Terms

The process of combining like terms is an essential skill in simplifying algebraic expressions. It involves identifying terms in an expression that have identical variable parts. These terms can be added or subtracted together, simplifying the expression into a more concise form.

• Like terms are those terms that contain the same variable raised to the same power. For example, in the expression \(x + 9x + 3\), both \(x\) and \(9x\) are like terms because they include the variable \(x\).
• To combine these terms, simply add or subtract their coefficients. In the given expression, \(x + 9x\) becomes \(10x\) because you add the coefficients (1 and 9) to get 10.

By consolidating like terms, you reduce the length and complexity of an expression, making it easier to work with in subsequent calculations.

Variables in Expressions

Variables are symbols, often letters like \(x\), \(y\), or \(z\), used in expressions to represent unknown or variable quantities. Understanding how to work with variables is fundamental in algebra.

• Variables stand in for values that may change or that are not yet known. For example, in \(x + 9x + 3\), the variable \(x\) indicates an unknown value.
• When simplifying expressions, recognizing that variable parts of terms need to match exactly in order to combine them is crucial. In "combining like terms," as previously discussed, this means identifying terms where the letter and its exponent are the same.

Recognizing how variables function within expressions allows you to manipulate and simplify those expressions effectively, paving the way for solving equations and understanding relationships within mathematics.

Algebraic Expressions

Algebraic expressions form the foundation of algebra, consisting of variables, numbers, and operations. Simplifying these expressions is a key skill that makes problem-solving manageable and efficient.

• An algebraic expression can consist of constants (like 3), variables (such as \(x\)), and arithmetic operations (such as addition and multiplication).
• Simplification often requires combining like terms and utilizing properties of operations (like distributive property) to rewrite the expression in a simpler form. For instance, from \(x + 9x + 3\) to \(10x + 3\).
• Simplified expressions help to solve equations or to evaluate the expression by substituting values for the variables.

Mastering the manipulation and simplification of algebraic expressions eases the path through more complex algebraic concepts, enabling deeper comprehension and problem-solving abilities in mathematics.