Problem 31 $$\text { Evaluate each expressi... [FREE SOLUTION]

Chapter 4: Problem 31

$$\text { Evaluate each expression if } x=-\frac{1}{3}, y=\frac{2}{5}, \text { and } z=\frac{5}{6} \text { . }$$ $$\frac{x}{z}$$

Short Answer

Expert verified

The expression $ \frac{x}{z} $ evaluates to $ -\frac{2}{5} $.

Step by step solution

Substitute the Values

Substitute the given values of the variables into the expression. Given that $ x = -\frac{1}{3} $ and $ z = \frac{5}{6} $, replace $ x $ and $ z $ in the expression $ \frac{x}{z} $ with these values. So, the expression becomes $ \frac{-\frac{1}{3}}{\frac{5}{6}} $.

Simplify the Fraction

To simplify $ \frac{-\frac{1}{3}}{\frac{5}{6}} $, multiply the numerator by the reciprocal of the denominator: $ -\frac{1}{3} \times \frac{6}{5} $.

Perform the Multiplication

Multiply the fractions: $ -\frac{1}{3} \times \frac{6}{5} = -\frac{6}{15} $.

Simplify the Resulting Fraction

Simplify $ -\frac{6}{15} $ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This gives $ -\frac{2}{5} $.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Variable Substitution

Variable substitution is a crucial technique in algebra that helps us to evaluate expressions with specific variable values. In our given exercise, we have expressions with variables $ x $ and $ z $, which need to be replaced by their designated values.

Substitution involves straightforward steps:

Identify the variable and its given value.
Replace the variable in the expression with this value.
This transforms the original algebraic expression into a numerical expression that can easily be evaluated.

For example, given the expression $ \frac{x}{z} $ and the values $ x = -\frac{1}{3} $ and $ z = \frac{5}{6} $, we substitute to get $ \frac{-\frac{1}{3}}{\frac{5}{6}} $. This step paves the way for further simplification.

Multiplication of Fractions

The multiplication of fractions is a process where we multiply the numerators together and the denominators together. This method is important when simplifying complex fractions, like having a fraction over a fraction.

Here's how it works:

Take the fractions involved: for example, $-\frac{1}{3}$ and the reciprocal of $\frac{5}{6}$, which is $\frac{6}{5}$.
Multiply the numerators: $-1 \times 6 = -6$.
Multiply the denominators: $3 \times 5 = 15$.

So, multiplying $-\frac{1}{3}$ by $\frac{6}{5}$ gives $-\frac{6}{15}$. This fraction results from the multiplication of each part, sticking strictly to multiplying straight across both the top and bottom.

Reciprocal

Understanding the concept of reciprocals is vital for fraction division. The reciprocal of a fraction is simply the inverse, meaning you switch the numerator and the denominator.

This process allows you to convert a division problem into a multiplication one, which can be more straightforward to solve:

For instance, consider the fraction $\frac{5}{6}$, its reciprocal is $\frac{6}{5}$.
To divide by a fraction like $\frac{5}{6}$, multiply by its reciprocal: instead of dividing, multiply by $\frac{6}{5}$.

Employing reciprocals transforms the division of $-\frac{1}{3} \div \frac{5}{6}$ into a multiplication $-\frac{1}{3} \times \frac{6}{5}$, thus simplifying the fraction subtraction and ensuring easier computations.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD is crucial in simplifying a fraction to its lowest terms.

To simplify a fraction:

Identify the numerators and denominators after multiplication, which, in our exercise, are -6 and 15, respectively.
Determine the GCD of these numbers. For -6 and 15, the GCD is 3.
Divide both the numerator and the denominator by this GCD.

Thus, $-\frac{6}{15}$ simplifies to $-\frac{2}{5}$ because both parts are divisible by their GCD, helping us achieve a neat and simplified fraction.

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91影视

$$\text { Evaluate each expression if } x=-\frac{1}{3}, y=\frac{2}{5}, \text { and } z=\frac{5}{6} \text { . }$$ $$\frac{x}{z}$$

Short Answer

Step by step solution

Substitute the Values

Simplify the Fraction

Perform the Multiplication

Simplify the Resulting Fraction

Key Concepts

Variable Substitution

Multiplication of Fractions

Reciprocal

Greatest Common Divisor

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