Question

Find the value of each expression if \(x=2\) and \(y=-1\). \(-3 x^{-1} y\)

Short answer

\( \frac{3}{2} \)

Step-by-step solution

Substitute the values of x and y

Replace the variables in the expression with the given values. Here, we have: \[ -3 x^{-1} y \] Given: \(x = 2\) and \(y = -1\), so substitute these into the expression: \[ -3 (2)^{-1} (-1) \]

Simplify the expression

Evaluate the exponent and multiplication. First, calculate \(2^{-1}\): \[ 2^{-1} = \frac{1}{2} \] Now, substitute this back into the expression: \[ -3 \left( \frac{1}{2} \right) (-1) \]

Multiply the fractions

Perform the multiplication: \[ -3 \cdot \frac{1}{2} \cdot (-1) \] First, calculate \( -3 \cdot \frac{1}{2} \): \[ -3 \cdot \frac{1}{2} = -\frac{3}{2} \] Then, multiply by \(-1\): \[ -\frac{3}{2} \cdot (-1) = \frac{3}{2} \]

Key concepts

Substitution

Substitution is a fundamental concept in algebra where you replace variables with given values. This step is crucial for evaluating expressions correctly. For example, in our exercise, we were given the values of x and y as 2 and -1, respectively.
We substituted these values into the expression \[-3 x^{-1} y\] to obtain:\[-3 (2)^{-1} (-1)\]
This step sets the groundwork for solving the rest of the problem. It helps us transform the algebraic expression into a numerical one, making it easier to work with.

Negative Exponents

Negative exponents often confuse students, but they are simpler than they appear. A negative exponent indicates that we take the reciprocal of the base. For instance, in our exercise, we had to evaluate \(2^{-1}\). This translates to:\[2^{-1} = \frac{1}{2}\]
Understanding this rule is essential for correctly simplifying expressions involving negative exponents. Whenever you see a negative exponent, remember to flip the base to its reciprocal, and then proceed with any other operations.

Multiplying Fractions

Multiplying fractions can be tricky, but breaking it down step-by-step makes it manageable. In our solution, we had to multiply several terms:

• First, we multiplied \(-3 \times \frac{1}{2}\), which simplifies to \(-\frac{3}{2}\).
• Next, we multiplied this result by \(-1\) to get \(\frac{3}{2}\).

Remember to follow these steps:
1. Simplify each fraction, if possible.
2. Multiply the numerators together.
3. Multiply the denominators together.
4. Simplify the resulting fraction, if needed.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing arithmetic operations. It's the final step to reach the solution. In this case, after substituting values and dealing with the negative exponent, we simplified the expression.
We first calculated \-3 \times \frac{1}{2}\, which gave us \- \frac{3}{2}\.
Then, we multiplied by \(-1\), which simplified the expression to \(\frac{3}{2}\).
Always follow the order of operations (PEMDAS/BODMAS) and simplify step-by-step for accurate results.