Problem 102 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(\frac{11-3 a^{2}}{y}\)

Step-by-Step Solution

Answer

4

Step 1 - Identify the values of the variables

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

Step 2 - Substitute the values into the expression

Substitute the values of \(a\) and \(y\) into the given expression \(\frac{11 - 3a^2}{y}\).This gives us: \(\frac{11 - 3(3)^2}{-4}\).

Step 3 - Simplify the expression inside the numerator

First, simplify the exponent and multiplication: \(3^2 = 9\).Then multiply: \(3 \cdot 9 = 27\).So, the expression becomes: \(\frac{11 - 27}{-4}\).

Step 4 - Compute the numerator

Subtract 27 from 11: \(11 - 27 = -16\).Now the expression is: \(\frac{-16}{-4}\).

Step 5 - Divide the numerator by the denominator

Divide \(-16\) by \(-4\): \(\frac{-16}{-4} = 4\).

Key Concepts

Substitution

Substitution is the process of replacing variables in an algebraic expression with given numerical values. When solving the expression \(\frac{11-3 a^{2}}{y}\), we substitute the values for the variables provided: \(a=3\) and \(y=-4\). This gives us \(\frac{11-3(3)^2}{-4}\). It’s essential to substitute the values correctly to ensure the rest of the steps are accurate.

Simplifying Expressions

Simplifying expressions involves performing operations within the expression to reduce it to its simplest form. After substituting the variables, our expression is \(\frac{11-3(3)^2}{-4}\). First, we handle the exponentiation and multiplication: calculate \((3)^2=9\), then multiply it by 3 to get 27. This simplifies our expression to \(\frac{11-27}{-4}\). Next, subtract 27 from 11, resulting in \(-16\). Now our expression is \(\frac{-16}{-4}\). By simplifying step-by-step, it becomes much easier to evaluate.

Exponentiation

Exponentiation refers to the operation of raising a number to the power of another number. In the given expression, we encounter \(3^2\). This means raising 3 to the power of 2, which equals 9. Understanding exponentiation is crucial because it establishes the foundation for simplifying the rest of the expression accurately.

Division of Numbers

Division splits a number into equal parts. In our problem, we need to divide \(-16\) by \(-4\). Dividing two negative numbers results in a positive number: \[\frac{-16}{-4}=4\]. Mastering division, especially with negative values, is essential for solving algebraic expressions correctly. Remember, the first number is the numerator and the second is the denominator. This clear distinction helps prevent common mistakes.