Problem 114 - Prealgebra and Introductory Algebra

Question

Evaluate the expression \(x y\) for the given values of \(x\) and \(y.\) $$x=-\frac{3}{10}, y=-35$$

Step-by-Step Solution

Answer

The value of the expression \(x y\) for \(x=-\frac{3}{10}, y=-35\) is 10.5.

Step 1: Substitute the variables

Replace \(x\) with \(-\frac{3}{10}\) and \(y\) with \(-35\) in the expression \(x y\). This yields \(-\frac{3}{10} \times -35\).

Step 2: Perform multiplication

Multiply \(-\frac{3}{10}\) by \(-35\). The minus signs cancel out to give a positive result because the product of two negative numbers is a positive number. So, this would be \(\frac{3}{10} \times 35 = 10.5\).

Key Concepts

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it's actually quite straightforward once you get the hang of it. When you multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. For example, if you have \(-\frac{3}{10}\), it means you are multiplying -3 by -35, while also considering the denominator.

In our given problem, after substituting the values, we have \(-\frac{3}{10}\) times \(-35\). Here, it helps to interpret -35 as \(\frac{-35}{1}\) so that both numbers are visually consistent as fractions. Once you align both terms, proceed by multiplying the numerators:

• -3 multiplied by -35 (minus sign is important here as it affects the outcome).
• 10 multiplied by 1.

This simplifies to \(\frac{105}{10}\), further calculated down to 10.5.

Remember, handling the numerators and denominators separately keeps your calculations clear and organized. It also ensures that even tricky problems like this one are handled with ease.

Negative Numbers

Handling negative numbers can appear confusing until you understand their basic operations. A key rule to remember is that when multiplying two negative numbers, the result is always a positive number.

Imagine negative numbers simply as directions on a number line: moving opposite a negative direction brings you into the positive. So, in our example:

• We have \(-\frac{3}{10}\) times \(-35\).
• The two negatives (\(-)(-35\)) interact multiplicatively to generate a positive product because 'going back' twice lands you forward.

Thus, our calculation switches from negative to positive, which results in just \(\frac{3}{10}\) times 35, leading to a positive 10.5. This concept helps keep operations logical and consistent, even when working with various mathematical contexts.

Substituting Variables

Substituting variables is like swapping placeholders with actual numerical values, which allows you to compute a clear result. It’s a foundational skill in algebra and mathematics generally. First, ensure you recognize what each variable represents in a given expression or equation.

In our problem, we start with the expression \(xy\), where \(x = -\frac{3}{10}\) and \(y = -35\). By substituting:

• Replace \(x\) with \(-\frac{3}{10}\).
• Replace \(y\) with \(-35\).

The expression then transforms into \(-\frac{3}{10} \times -35\).

This step is crucial for translating the algebraic expressions into numerical computations, allowing you to perform calculations with known integers or decimals. Be sure to substitute accurately, keeping in line with the operation signs attached to each variable, as this preserves the mathematical intent of the problem.