Question

Evaluate each expression if \(x=-2\) and \(y=3\). \(\frac{2 x}{x-y}\)

Short answer

\(\frac{4}{5}\)

Step-by-step solution

Substitute the variable values

Start by substituting the given values of the variables into the expression. Given that \(x = -2\) and \(y = 3\), we can replace \(x\) and \(y\) in the expression \(\frac{2 x}{x-y}\). This gives us:\[\frac{2(-2)}{-2-3}\]

Simplify the numerator

Simplify the numerator by multiplying 2 and \(x\). Since \(x = -2\), the numerator is:\[2 \cdot (-2) = -4\]

Simplify the denominator

Simplify the denominator by performing the subtraction \(x - y\). Since \(x = -2\) and \(y = 3\), the denominator is:\[-2 - 3 = -5\]

Divide the numerator by the denominator

Finally, divide the simplified numerator by the simplified denominator:\[\frac{-4}{-5} = \frac{4}{5}\]

Key concepts

Substitution in Algebra

Substitution is a basic technique in algebra where you replace variables with their given values. This process makes it possible to evaluate expressions and find specific numerical results. For instance, if you are given that \( x = -2 \) and \( y = 3 \), you can substitute these values into any algebraic expression involving \( x \) and \( y \). In our example, we substituted these values into the expression \( \frac{2x}{x-y} \):

First, substitute to get:\( \frac{2(-2)}{-2-3} \).

This step is crucial because it translates the abstract expression into a more concrete format that can be simplified using arithmetic operations. Always ensure the substitution is correct before moving on to simplification.

Simplifying Numerators and Denominators

After substituting the values, the next step involves simplifying the numerator and the denominator separately. Simplification often includes performing basic arithmetic operations like multiplication, addition, or subtraction.

In our example, the expression after substitution was \( \frac{2(-2)}{-2-3} \). Here is how you simplify each part:

• Simplify the numerator: Multiply \( 2 \) and \( -2 \) to get \( -4 \).
• Simplify the denominator: Subtract \( 2 \) from \( 3 \) to get \( -5 \).

So, the expression now looks like this: \( \frac{-4}{-5} \).Breaking down these steps ensures that any errors can be easily spotted and corrected. Make sure you handle negative signs carefully during simplification.

Basic Arithmetic Operations

The final step involves performing basic arithmetic operations, specifically division in this example. After simplifying the numerator and denominator, you often need to divide the two results to complete the evaluation.

From our simplified expression, which was \( \frac{-4}{-5} \), we divide \( -4 \) by \( -5 \). When you divide two negative numbers, the result is positive, so:
\( \frac{-4}{-5} = \frac{4}{5} \)

Basic arithmetic operations like addition, subtraction, multiplication, and division form the foundation of algebra. Mastering these operations will make solving more complex expressions much easier. Always check your work as you go to avoid mistakes, and use parentheses where needed to clarify order of operations.