Problem 96 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(\left(\frac{5}{6} x+\frac{3}{2} y\right)\left(-\frac{1}{3} a\right)\)

Step-by-Step Solution

Answer

The expression evaluates to 1.

Step 1: Identify the Variables

Given values are: \(x = 6\), \(y = -4\), and \(a = 3\).

Step 2: Substitute the Variables

Substitute \(x\), \(y\), and \(a\) into the expression: \[ \frac{5}{6}x + \frac{3}{2}y \rightarrow \frac{5}{6}(6) + \frac{3}{2}(-4) \ \text{and} \ -\frac{1}{3}a \rightarrow -\frac{1}{3}(3) \ \text{So the expression becomes:} \ \bigg( \frac{5}{6}(6) + \frac{3}{2}(-4) \bigg)\bigg( -\frac{1}{3}(3) \bigg)\]

Step 3: Simplify Inside the Parentheses

Calculate each part inside the parentheses: \[ \frac{5}{6}(6) = 5 \ \frac{3}{2}(-4) = -6 \ \bigg(5 + (-6)\bigg) \bigg( -1 \bigg) \ -1 \times -1\]

Step 4: Perform Final Calculation

Multiply the simplified results: \[ (-1) \times (-1) = 1 \]

Key Concepts

Substitution

When solving an algebraic expression for specific values of variables, the first step is **substitution**. This means replacing each variable in the expression with the given numerical value. In our problem, we were given the values for three variables:

• \(x = 6\)
• \(y = -4\)
• \(a = 3\)

Substituting these into the expression \( \left(\frac{5}{6}x + \frac{3}{2}y\right)\left(-\frac{1}{3}a\right)\) involves these specific replacements:

• \(\frac{5}{6}x \rightarrow \frac{5}{6}(6)\)
• \(\frac{3}{2}y \rightarrow \frac{3}{2}(-4)\)
• \(-\frac{1}{3}a \rightarrow -\frac{1}{3}(3)\)

After applying these substitutions, our expression turns into:
\[ \left(\frac{5}{6}(6) + \frac{3}{2}(-4)\right)\left(-\frac{1}{3}(3)\right) \] which sets the stage for the next steps.

Simplification

The next core concept is **simplification**. Simplification involves performing the arithmetic operations inside the expressions to reduce them to simpler forms. Let's break this down step by step.
First, simplify the parts inside the parentheses:

• Calculate \(\frac{5}{6}(6)\):
  This simplifies to 5, because multiplying by 6 cancels out the denominator.
• Calculate \(\frac{3}{2}(-4)\):
  This simplifies to -6, because \(\frac{3}{2}\) times \(-4\) equals -6.

Combining these simplifications, we get:
\[ 5 + (-6) = -1 \]
Now for the outer part of our expression:
\(-\frac{1}{3}(3)\) simplifies to -1. These simplified values make our final step easier.

Multiplication of Fractions

For multiplication of fractions, there's a special rule: simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. If your fraction is multiplied by an integer, treat the integer as a fraction with a denominator of 1.
Example from the problem:
\(-\frac{1}{3}(3)\) can be viewed as:
\(-\frac{1}{3} \times \frac{3}{1} = -\frac{1 \times 3}{3 \times 1} = -1\)
Similarly, when substituting into the expression \(\frac{3}{2}(-4)\), you treat -4 as \(\frac{-4}{1}\).
This process helps clarify why our simplifications work the way they do. Every fraction multiplication adheres to these rules to give accurate results.

Negative Numbers

**Negative numbers** follow specific rules during arithmetic operations which aid in simplification.
Consider the rules when dealing with negative signs:

• **Multiplication or division of two negative numbers yields a positive result.** For instance, \(-1 \times -1 = 1\).
• **Multiplication or division of a positive and a negative number results in a negative.** For example, \(5 \times -1 = -5\).

In our expression, when we concluded with \(-1 \times -1\), we used the first rule: two negatives multiply to a positive:
\[(-1) \times (-1) = 1\] These rules are fundamental to avoid errors and ensure accuracy.