Problem 31 - Algebra 1

Question

Evaluate the expression. $$ \frac{-6}{q}-2 r \text { when } q=2 \text { and } r=11 $$

Step-by-Step Solution

Answer

-25

Step 1: Substituting the Given Values

To begin, substitute the given values of \(q\) and \(r\) into the expression. This gives you: \(-\frac{6}{2} - 2(11)\).

Step 2: Simplifying the Expression

We then simplify each term in the expression individually. We know that \(-\frac{6}{2}= -3\) and \(2(11) = 22\). Thus, the expression simplifies to \(-3 - 22\).

Step 3: Performing the Subtraction

Finally, subtract \(22\) from \(-3\). This gives us a final value of \(-3 - 22 = -25\).

Key Concepts

Substituting Variables

When faced with algebraic expressions, substituting variables is akin to replacing a placeholder with its actual value. This process is crucial to simplifying expressions and finding solutions to algebraic problems. For instance, consider the expression \(\frac{-6}{q}-2 r\) where you're given \(q=2\) and \(r=11\).

Substitution means that wherever you see the variable \(q\), you replace it with 2, and wherever you see \(r\), you replace it with 11. It's a direct swap. So, the original expression becomes \(\frac{-6}{2}-2(11)\). Remember, when substituting, it's crucial to maintain the integrity of the original expression. Each variable should be replaced with its given value with careful attention to the operators (like addition, subtraction, multiplication, and division) that surround them.

Evaluating Expressions

Evaluating expressions is the systematic process of completing operations to find the value of an expression. After substituting variables with given values, you look at the operations required. The expression \(\frac{-6}{2}-2(11)\) contains two operations: division and multiplication, followed by subtraction. Begin with division and multiplication because of the order of operations (often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

So, \(\frac{-6}{2}\) evaluates to -3, and \(2(11)\) evaluates to 22. Breaking expressions down into these simpler steps makes it more manageable. Only after evaluating each operation do you move to combining these results, which in this case involves the final operation of subtraction.

Performing Arithmetic Operations

Performing arithmetic operations is the fundamental skill required for simplifying algebraic expressions. After evaluating the individual parts of the expression \(\frac{-6}{2}-2(11)\) and finding that they are -3 and 22, we then focus on the remaining subtraction. Remember, there's a hidden order here: Always work from left to right.

Subtracting 22 from -3 may be initially confusing, but visualize it as moving further left on the number line. Starting at -3, moving 22 places left lands you at -25. Thus, \( -3 - 22 \) equals \( -25 \). Mastering these arithmetic operations—addition, subtraction, multiplication, and division—is crucial since algebra builds on these core mathematical principles, and they are tools used in every aspect of solving algebraic expressions.