Problem 134 - Elementary and Intermediate Algebra

Question

Evaluate each expression for \(x=-3\) and \(y=-5\) $$ \frac{2 y+1}{x}-x $$

Step-by-Step Solution

Answer

The evaluated expression is 6.

Step 1: Substitute the Values

Start by substituting the given values of \(x = -3\) and \(y = -5\) into the expression \(\frac{2y+1}{x} - x\). This results in \(\frac{2(-5) + 1}{-3} - (-3)\).

Step 2: Simplify the Numerator

Calculate the expression \(2(-5) + 1\). Multiply 2 by -5 to get -10 and then add 1 to get -9. So, the expression becomes \(\frac{-9}{-3} - (-3)\).

Step 3: Evaluate the Division

Divide the numerator by the denominator: \(\frac{-9}{-3} = 3\) since dividing a negative by a negative gives a positive result.

Step 4: Simplify the Expression

Substitute the division result back into the expression: \(3 - (-3)\). Simplifying \(3 + 3\), because subtracting a negative is equivalent to adding, results in \(6\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra. It involves replacing variables in an expression with given numerical values. This is a crucial step when evaluating expressions, especially when you are given specific values for the variables involved. Substitution transforms an algebraic expression into a numerical expression, making it easier to handle and solve. To effectively use the substitution method:

• Identify the variables in the expression that you need to substitute with numbers.
• Carefully replace each variable with its corresponding value given in the problem.
• Keep the arithmetic operations intact, ensuring that the correct order of operations is applied post substitution.

In the provided problem, we started by substituting \( x = -3 \) and \( y = -5 \) into the expression \( \frac{2y+1}{x} - x \). This initial step sets the stage for further manipulation and simplification of the expression.

Simplifying Expressions

Simplifying expressions involves performing arithmetic operations and reducing the expression to a simpler form. After substituting values, it's important to simplify to reveal the expression's basic components.Here are steps to simplify efficiently:

• Perform operations inside parentheses first.
• Combine like terms whenever possible.
• Simplify fractions by carrying out any division.

In the example, after substituting values, we calculated \( 2(-5) + 1 \) to simplify the numerator. This process involved multiplying first \( 2 \times -5 = -10 \) and then adding \( 1 \) to get \( -9 \). Simplification then involved dividing the fraction: \( \frac{-9}{-3} \), resulting in \( 3 \).

Evaluating Expressions

Evaluating expressions is the final step in solving mathematical problems involving variables. Once values are substituted and the expression is simplified, evaluating allows us to find the precise numerical result.To evaluate correctly:

• Ensure that all operations, including division and subtraction, are completed correctly.
• Pay attention to signs; negative and positive numbers should be handled carefully.
• Use the order of operations (also known as PEMDAS/BODMAS) - Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In the example, evaluating involved calculating the result of \( 3 - (-3) \). By recognizing that subtracting a negative number is equivalent to addition, the final step resulted in \( 3 + 3 \), which equals \( 6 \). Through careful evaluation, we arrived at the final answer.