Problem 73 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression and simplify if possible. Let \(x\) represent the unknown number. Double a number minus the sum of the number and ten

Step-by-Step Solution

Answer

The simplified algebraic expression is \(x - 10\).

Step 1: Translate the Phrase to Algebraic Expression

The phrase 'double a number' can be translated to the expression \(2x\). The phrase 'the sum of the number and ten' can be translated to \(x + 10\). Therefore, the expression for the entire phrase 'double a number minus the sum of the number and ten' becomes \(2x - (x + 10)\).

Step 2: Distribute the Negative Sign

Evaluate \(2x - (x + 10)\) by distributing the negative sign through the terms in the parentheses. This gives us \(2x - x - 10\).

Step 3: Simplify the Expression

Combine like terms in the expression \(2x - x - 10\). This simplifies to \(x - 10\).

Key Concepts

Simplifying Expressions

Simplifying expressions in algebra is all about making them easier to work with and understand. When you simplify an expression, you're essentially looking to reduce it to its most basic form. This involves the elimination of any unnecessary complexity by performing operations and reducing like terms. Take for instance the expression \(2x - (x + 10)\). At first glance, it looks quite daunting. But by cleverly distributing and combining, you can simplify it. This process is important because it helps in solving equations more effectively down the line. When you simplify algebraic expressions, you're ensuring you're only dealing with the numbers and variables you need—nothing more and nothing less. In this way, simplified expressions become a foundational step in solving more complex algebraic problems.

Combining Like Terms

Combining like terms is a crucial step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power, though the coefficients, or numbers in front of the variables, may differ. When you combine these terms, you are essentially grouping them together to form a single term.In the expression \(2x - x - 10\), for instance, both \(2x\) and \(x\) are like terms because they both involve the variable \(x\). You can combine them by subtracting the coefficient of \(x\) from the coefficient of \(2x\). So, \(2x - x\) simplifies to \(x\). The \(-10\) remains unchanged as it is not like the \(x\) terms. Combining like terms allows for the expression to be more compact, making further algebraic manipulation simpler.

Distributive Property

The distributive property is a key tool in algebra that allows us to remove parentheses by distributing a factor across terms within the parentheses. It's often expressed as \(a(b + c) = ab + ac\). In our example, consider the expression \(2x - (x + 10)\). The minus sign in front of the parentheses needs to be distributed across each term inside. This means \(-1\) is multiplied by each of the terms inside the parenthesis: \(-1 \times x\) and \(-1 \times 10\), rewriting the expression as \(2x - x - 10\). The distributive property helps in breaking down complex expressions into simpler pieces, making it easier to combine like terms to further simplify the expression. Understanding and applying this property correctly is essential for solving algebraic expressions accurately.