Problem 50 - Elementary Algebra Within Reach

Question

In Exercises \(45-54\), evaluate the algebraic expression for the given values of the variables. If it is not possible, state the reason. \(\frac{5 x}{y-3}\) (a) \(x=2, y=4\) (b) \(x=2, y=3\)

Step-by-Step Solution

Answer

(a) 10, (b) undefined due to division by zero

Step 1: Substitute the given values for x and y (case a)

We first replace x by 2 and y by 4 in the expression, yielding \(\frac{5 * 2}{4 - 3}\),which simplifies to \(\frac{10}{1}\).

Step 2: Perform the division (case a)

Perform the division gives us the final result, 10.

Step 3: Substitute the given values for x and y (case b)

Next, we substitute x by 2 and y by 3 in the expression, yielding \(\frac{5 * 2}{3 - 3}\),which simplifies to \(\frac{10}{0}\).

Step 4: Discuss the undefined result (case b)

As division by zero is undefined in mathematics, we conclude that the value of the algebraic expression is not defined for x = 2 and y = 3.

Key Concepts

Substitution

Substitution is a fundamental technique in algebra that allows you to evaluate expressions by replacing variables with given values. When you come across an algebraic expression like \(\frac{5x}{y-3}\), and are asked to evaluate it for specific values, your first step is to substitute these values into the equation. This involves:

• Identifying the variables present in the expression.
• Replacing each variable with the numbers provided in the problem.

For example, in our expression \(\frac{5x}{y-3}\), if you substitute \(x=2\) and \(y=4\), you replace each instance of \(x\) and \(y\) in the expression to form \(\frac{5 \cdot 2}{4 - 3}\). Through substitution, the expression transforms from one involving variables to one involving only numbers, which you can then solve using arithmetic operations.
It's crucial to ensure that you substitute carefully to avoid simple mistakes, as these can change the problem entirely.

Undefined Expressions

In algebra, you might encounter situations where evaluating an expression leads to an undefined result. One classic case is division by zero. This is a crucial point to understand because mathematically, dividing by zero makes the expression meaningless or undefined.

For the expression \(\frac{5x}{y-3}\), when you substitute \(x=2\) and \(y=3\), the denominator becomes zero: \(3 - 3 = 0\). As a result:

• You try to evaluate \(\frac{10}{0}\).
• Since division by zero is undefined, the entire expression has no defined value.

Understanding undefined expressions helps prevent logical errors in mathematics and highlights the importance of scrutinizing expressions before performing operations.

Algebraic Fractions

Algebraic fractions are expressions that involve fractions with variables in the numerator, the denominator, or both. They resemble numerical fractions, but with the added complexity of variables, which means their values can change depending on the variables they contain.

For example, in \(\frac{5x}{y-3}\), \(5x\) is the numerator, and \(y-3\) is the denominator. Working with algebraic fractions involves steps like:

• Simplifying the fractions by canceling common factors.
• Substituting given values to solve.

However, you must always remember to check if the denominator equals zero after substitution, as this would make the expression undefined.
Knowing how to handle algebraic fractions is vital as they frequently appear in equations and inequalities in algebra, offering both challenges and opportunities for problem-solving.