Problem 104 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(\frac{x y+8 a}{x-6}\)

Step-by-Step Solution

Answer

The expression is undefined.

Step 1 - Substitute the values

Replace the variables in the expression \(\frac{x y+8 a}{x-6}\) with the given values: \(x = 6, y = -4, a = 3.\). The expression becomes: \(\frac{6(-4) + 8(3)}{6-6}\)

Step 2 - Simplify the numerator

Calculate the value of the numerator. First, multiply the values inside the parentheses: \(6 \times (-4) = -24\) and \(8 \times 3 = 24\). Then, add these results together inside the numerator: \(-24 + 24 = 0\)

Step 3 - Simplify the denominator

Calculate the value of the denominator. Subtract the given values: \(6 - 6 = 0\)

Step 4 - Evaluate the expression

Substitute the simplified numerator and denominator into the fraction: \(\frac{0}{0}\). Therefore, the expression is undefined because division by zero is not possible.

Key Concepts

Substitution

When evaluating expressions, the first key step is substitution. Substitution means replacing variables with given values. To master substitution, follow these steps:

Identify the variables within the expression.
Replace each variable with its provided value.
For example, given the expression \(\frac{x y + 8 a}{x - 6}\) and values \(x = 6, y = -4, a = 3\), substitution transforms it into \(\frac{6(-4) + 8(3)}{6-6}\).

Always double-check the values substituted to ensure they match the given problem. This sets the foundation for all further calculations.

Simplification

Simplifying an expression involves breaking it down into its simplest form. This can include reducing fractions, combining like terms, and performing arithmetic operations. Let’s go through the simplification steps used in the example:

Start by multiplying the substituted values: \(6 \times (-4) = -24\) and \(8 \times 3 = 24\).
Combine the results: \(-24 + 24 = 0\).
Simplify the denominator by computing: \(6 - 6 = 0\).

After these steps, the expression simplifies to \(\frac{0}{0}\). It is essential to understand each operation and simplify systematically to avoid mistakes.

Undefined Expression

An undefined expression occurs when the calculation leads to a result that is not mathematically valid. The most common case is division by zero.

In our example, after simplification, we are left with \(\frac{0}{0}\). Division by zero is undefined because it does not provide a meaningful value in mathematics. This makes the entire expression undefined.

Always watch out for division by zero when simplifying expressions. If you encounter such a situation, clearly state that the expression is undefined.

Understanding why an expression is undefined helps in recognizing and addressing such cases in other problems.