Problem 41 - Algebra 1

Question

Evaluate the expression when \(x=-3\) and \(y=6 .\) $$\frac{3 x}{x+y}$$

Step-by-Step Solution

Answer

The evaluation of the expression is -3.

Step 1: Substitute the given values

Replace each variable in the expression with the given values. This gives us: \( \frac{3(-3)}{-3+6} \).

Step 2: Simplify the numerator and denominator

Execute multiplication in the numerator and the addition in the denominator: -9 in the numerator and 3 in the denominator.

Step 3: Execute the division

Execute the division operation by dividing the numerator by the denominator: \(\frac{-9}{3}\).

Key Concepts

Substitution

Substitution in algebra is all about replacing variables with actual numbers. Imagine working with an equation where you need to evaluate something like \(\frac{3x}{x+y}\). Once you're told \(x = -3\) and \(y = 6\), your job becomes straightforward: insert these numbers into the expression.

Think of each variable as a placeholder. Here, \(x\) and \(y\) are placeholders. Replace \(x\) with -3 and \(y\) with 6. This step is crucial because without it, the algebraic expression would remain abstract. The substitution gives us a concrete form: \(\frac{3(-3)}{-3+6}\). By doing this, you've set the stage for the next steps of solving your expression.

Numerical Evaluation

Numerical evaluation involves performing arithmetic operations to simplify expressions. Once you've substituted values for variables, the next step is to calculate. Let's consider the expression \(\frac{3(-3)}{-3+6}\).

First, address the numerator: you multiply 3 by -3, yielding -9. Multiplication comes first, according to basic arithmetic rules. Next, manage the denominator by adding -3 and 6, resulting in 3. Simplifying these components one by one helps avoid mistakes.

• The numerator, calculated as \(3 \times -3\), gives -9.
• The denominator simplifies to \(-3 + 6\), resulting in 3.

You've turned a complex expression into simple numbers, making the final step straightforward.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. After substitution and numerical evaluation, simplify the fraction by executing the division. Consider the expression \(\frac{-9}{3}\).

This fraction represents a division. You are essentially dividing the number -9 by 3. Ask yourself how many times 3 fits into -9, which is -3 times. The sign of the result is negative because a negative number divided by a positive number is negative.

• Divide the absolute values: \(9 \div 3 = 3\).
• Apply the sign rule: negative divided by positive yields negative (-3).

The final result is -3, as the fraction \(\frac{-9}{3}\) simplifies completely.