Problem 83 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to -12.

Step 1: Substitute the Given Values

First, substitute the given values of \(x\), \(y\), and \(z\) into the expression \((3 + x) y\). Here, \(x = 3\) and \(y = -2\). So, substitute these into the expression: \((3 + 3) \cdot (-2)\).

Step 2: Simplify Inside the Parentheses

Now, simplify the expression inside the parentheses. Add \(3\) and \(3\) to get \(6\). Your expression is now \(6 \cdot (-2)\).

Step 3: Multiply the Remaining Numbers

Multiply the simplified number by \(y\). Multiply \(6\) by \(-2\) to get \(-12\). So, the expression evaluates to \(-12\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra that involves replacing variables in an expression with their corresponding numerical values.
This allows us to evaluate the expression or solve equations.
For example, when given an expression like \( (3+x)y \)\ and values for the variables, substitute the given numbers directly for the appropriate letters.

• In our case, if \( x = 3\) and \( y = -2 \), you would replace \( x \) and \( y \) with these numbers.
• This transforms the expression into \((3 + 3) \cdot (-2)\).

Performing substitution simplifies the process of evaluating expressions and equations, providing a clear numerical outcome. It is an essential step for further manipulations like addition, multiplication, and other arithmetic operations that might follow.

Simplifying Expressions

Simplifying expressions is all about breaking down complex parts of a mathematical expression into simpler components.
This makes the expression easier to work with.
In the context of our exercise, after substituting, you end up with an expression such as \((3 + 3) \cdot (-2)\).

• The first task is to simplify inside the parentheses.
• Add the numbers within: \(3 + 3 = 6\).

Once simplification within parentheses is complete, you're left with an expression that's much easier to evaluate.Simplifying reduces mistakes and clarifies the problem for further operations, such as multiplication.

Multiplication in Algebra

Multiplication in algebra involves scaling numbers and connecting terms through the arithmetic operation of multiplying.Following simplification, the next step involves applying multiplication.
With our expression, \(6 \cdot (-2)\), we take the simplified result **6** and multiply by **-2**.

• Multiplication with a negative number follows the rule that a positive number times a negative number equals a negative result.
• Hence, \(6 \cdot (-2) = -12\).

In algebra, understanding how multiplication affects signs and values is critical. It simplifies and finalizes the evaluation of expressions, clarifying what the expression equates to in numbers. Multiplying both simple and complex expressions solidifies your grasp of algebraic rules and properties.