Problem 76 - Algebra 1: Concepts and Skills

Question

Which expression is equivalent to \((5-x)(-17) ?\) $$(A) 5-17 x$$ $$(B) 5+17 x$$ $$(C) -85-17 x$$ $$(D) -85+17 x$$

Step-by-Step Solution

Answer

The expression equivalent to \((5-x)(-17)\) is \(-85 + 17x\), which corresponds to answer choice (D).

Step 1: Apply the Distributive Property

In the given expression \((5-x)(-17)\), use the distributive property by multiplying -17 with both terms inside the brackets separately. This gives: \(-17*5 - -17*x\)

Step 2: Simplify The Multiplication

Now, perform the multiplication for each term. This gives: \(-85 - -17x\)

Step 3: Simplify The Double Negative

Note the double negative '--' which can be simplified to a simple '+'. This gives: \(-85 + 17x\)

Key Concepts

Equivalent Expressions

Equivalent expressions are different algebraic statements that have the same value for all possible variable values. Think of them as different ways of saying the same thing, much like how '10 dimes' is equivalent to '1 dollar'.

In algebra, to determine if expressions are equivalent, one could perform operations such as addition, subtraction, multiplication, or division consistently across both expressions, or sometimes, factoring or expanding expressions. When evaluating the expression \(5-x)(-17)\), we aim to find its equivalent among the given options by employing the distributive property to simplify it. The correctness of equivalent expressions can be proven by reaching the same numerical result upon substituting values for the variables involved in the expressions.

Algebraic Multiplication

Algebraic multiplication involves multiplying variables and numbers, often requiring the use of the distributive property. This property states that for any numbers or expressions \(a\), \(b\), and \(c\), the equation \(a(b + c) = ab + ac\) holds true. In the context of the exercise, multiplication is carried out between a numeric value \( -17 \) and an algebraic expression \( (5 - x) \).

Using Distributive Property

The expression \( -17(5 - x) \) is distributed as \( -17 \cdot 5 + (-17) \cdot (-x) \) by multiplying -17 with each term inside the parentheses separately. This seminal algebraic skill allows us to crack open expressions and manipulate them into different, yet equivalent forms.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the process of reducing them into a simpler or more efficient form without changing their value. This involves combining like terms, factoring, expansion, and handling negatives and positives correctly.

After distributing and multiplying through the expression \( (5-x)(-17) \) as seen in the step by step solution, we encounter a double negative, which often confuses students. The rules of arithmetic teach us that two negatives make a positive, effectively turning \( - -17x \) into \( +17x \). Simplification is an essential skill in algebra, allowing us to express complex ideas in a more digestible and often more useful format.