Question

In Problems \(97-108\), find the value of each expression if \(x=2\) and \(y=-1\). \(2 x y^{-1}\)

Short answer

-4

Step-by-step solution

- Understand the Expression

Identify the given expression: the expression is \(2 x y^{-1}\). The values for the variables are \(x = 2\) and \(y = -1\).

- Substitute the Variables

Replace \(x\) and \(y\) in the expression with their given values: \(2 \cdot 2 \cdot (-1)^{-1}\).

- Simplify the Exponent

Simplify \((-1)^{-1}\). Recall that \(a^{-1} = \frac{1}{a}\), so \((-1)^{-1} = \frac{1}{-1}\). Therefore, the expression becomes \(2 \cdot 2 \cdot \frac{1}{-1} = 2 \cdot 2 \cdot -1\).

- Perform Multiplication

Multiply the numbers: \(2 \cdot 2 = 4\) and then \(4 \cdot -1 = -4\).

Key concepts

Variable Substitution

In algebra, variable substitution involves replacing variables in an expression with specific values. This is often one of the first steps in solving an algebraic problem.
Let's apply this to our exercise. We have the expression \(2xy^{-1}\) and we need to find its value when \(x = 2\) and \(y = -1\).

1. Identify the variables: \(x\) and \(y\).
2. Substitute the given values into the expression:
\(2xy^{-1} = 2 \times 2 \times (-1)^{-1}\).
By substituting the values, we transform the abstract expression into a numerical one, making it easier to solve.

Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal. This concept transforms an expression and simplifies calculations.
Some key points to remember about negative exponents:

• \(a^{-1} = \frac{1}{a}\): This shows that any number raised to the power of -1 is equal to its reciprocal.
• All negative exponents can be converted to positive exponents by moving the base to the denominator.

Applying this to our example, we have \((-1)^{-1}\). According to the rule, \((-1)^{-1} = \frac{1}{-1} = -1 \).
Incorporating this back into our expression gives: \(2 \times 2 \times -1\).
Understanding and applying negative exponents correctly can simplify many algebraic problems.

Multiplication of Integers

Multiplying integers is one of the basic arithmetic operations, but it can sometimes be tricky when negative numbers are involved.
Let's break it down step-by-step in our example:
1. Multiply the first two integers \(2 \times 2 = 4\).
2. Now, multiply the result by \(-1\): \(4 \times -1 = -4\).
Some important rules to remember while multiplying integers:

• A positive number multiplied by a positive number gives a positive result.
• A positive number multiplied by a negative number gives a negative result.
• A negative number multiplied by a negative number gives a positive result.

By understanding these rules, you'll be better prepared to handle more complex algebraic equations involving multiplication.