Problem 77 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ -y $$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Substitute the given value

You need to substitute the given value of the variable into the expression. Here, the expression is \(-y\). We know that the value of \(y\) is \(-2\). This gives:\[ -(-2) \]

Step 2: Simplify the expression

Apply the rule of double negatives in the expression. The negative sign before the brackets changes the sign of the number inside:\[ -(-2) = 2 \]The negative of a negative number is a positive number.

Key Concepts

Substitution in Algebra

Substitution in algebra involves replacing variables with their given numerical values. This process helps simplify expressions by providing specific numbers to work with. For example, in the exercise, we substituted the given value of \(y\) which is \(-2\) into the expression \(-y\). This step is crucial as it transforms an algebraic expression into a numerical form that can be further simplified.

The key steps in substitution are:

• Identify the variables in the expression.
• Replace each variable with its given value.
• Rewrite the expression with numerical values instead of variables.

By mastering substitution, you can tackle a wide range of algebraic problems, making equations and expressions easier to manage and solve.

Negative Numbers in Algebra

Dealing with negative numbers in algebra can sometimes be tricky, especially when they appear in expressions. Understanding how to handle negative numbers correctly is vital for solving algebraic expressions.

In the given exercise, we came across the expression \(-(-2)\). When dealing with double negatives, remember that two negative signs make a positive. So, \(-(-2)\) simplifies to 2.

Some tips for working with negative numbers are:

• A single negative sign inverts the sign of the number, making positive numbers negative and vice versa.
• A double negative inverts the sign again, returning the number to its original sign.
• Always pay close attention to where negative signs are placed in an expression to avoid errors.

Working with negative numbers becomes intuitive with practice, allowing you to solve algebraic expressions accurately.

Simplification of Expressions

Simplifying expressions in algebra is the process of rewriting them in their most concise and manageable form. This might involve combining like terms, applying arithmetic operations, or eliminating unnecessary elements like double negatives.

In our exercise, after substitution, we simplified \(-(-2)\) to 2. Here, we utilized the rule of double negatives to achieve simplification. Simplification makes complex expressions easier to understand and next steps, like solving equations, more straightforward.

Steps to simplify an expression include:

• Perform operations like addition, subtraction, multiplication, or division as needed.
• Apply algebraic rules such as those for negative numbers or powers appropriately.
• Simplify further by reducing fractions or eliminating redundant terms.

Simplification is a crucial skill in algebra, paving the way for solving more complex equations or systems.