Problem 95 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

-\(\frac{78}{25}\)

Step 1 - Substitute the values

First, substitute the given values for each variable in the expression. Given:\(x = 6, y = -4, a = 3\)The expression becomes \(\left(\frac{1}{3} (6)-\frac{4}{5} (-4)\right)\left(-\frac{1}{5} (3)\right)\)

Step 2 - Simplify inside the parentheses

Simplify the terms inside the parentheses.\(\frac{1}{3} (6) = 2\)\(\frac{4}{5} (-4) = -\frac{16}{5}\)Thus, the expression inside the parentheses is:\(2 - (-\frac{16}{5})\) which simplifies to \(2 + \frac{16}{5}\)

Step 3 - Combine the terms inside the parentheses

Combine the terms inside the parentheses to get a single fraction:\(2 = \frac{10}{5}\), so \(\frac{10}{5} + \frac{16}{5} = \frac{26}{5}\)The expression then becomes \(\frac{26}{5}\)

Step 4 - Multiply by the remaining term

Now multiply \(\frac{26}{5}\) by the remaining term on the outside of the parentheses:\(\frac{26}{5} \times -\frac{1}{5} (3) = -\frac{26}{5} (\frac{3}{5}) = -\frac{78}{25}\)

Key Concepts

Substitution in Algebra

Substitution in algebra is like putting puzzle pieces into the right slots. You replace the variables (letters) with their given values. In the expression we're working with, we're given:

• For variable x: 6
• For variable y: -4
• For variable a: 3

To substitute these values, you replace all instances of x, y, and a in the expression with 6, -4, and 3 respectively. So, the expression: a. Turns into: a. Substitution sets the stage before you can solve the expression.
It helps ensure you're working with numbers instead of letters, making the problem more straightforward.

Simplifying Fractions

Simplifying fractions is crucial to make calculations easier. It involves reducing fractions to their simplest form. In our problem:

We first handle each part individually.

For example:

Similarly, calculating gives us:

This part can be tricky, but breaking it down step-by-step helps. Simplification helps convert more complex fractions into simpler ones, making the algebraic manipulation easier.

Combining Like Terms

Combining like terms is like grouping similar items together. In algebra, it means adding or subtracting terms that have the same variable raised to the same power. In this problem, after substitution, we get: In this case:

Convert So, you're left with:

Combining these gives you a single fraction:
The simplified form helps you see the problem more clearly and makes it easier to handle the next steps.

Order of Operations

Order of operations dictates the sequence in which you solve parts of an expression. You can remember it with PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

First, solve inside the parentheses:

This means substituting and combining like terms:

Then the final step, multiplication:
Using the order of operations ensures you solve the expression correctly and systematically.