Problem 84 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The value of the expression is 0.

Step 1: Substitute the Values into the Expression

Replace the variables in the expression \((4+z) y\) with their given values. Here, \(z = -4\) and \(y = -2\). The expression becomes \((4 + (-4))(-2)\).

Step 2: Simplify Inside the Parentheses

Simplify the expression inside the parentheses. Calculate \(4 + (-4)\), which equals \(0\). Now the expression is \(0 \times (-2)\).

Step 3: Multiply the Simplified Terms

Multiply the result from Step 2 by \(y\). This means multiplying \(0\) by \(-2\). The result is \(0\) because anything multiplied by zero is zero.

Key Concepts

Understanding Variable Substitution

Variable substitution is like replacing placeholders in a formula with actual numbers. Let's say we have an expression with variables like \(x\), \(y\), or \(z\). To evaluate this expression when given certain values, we swap the variables with the numbers provided.
For example, if the expression is \((4 + z)y\) and we are told that \(z = -4\) and \(y = -2\), we substitute these into the expression.

• Start with \(4 + z\), replace \(z\) with \(-4\).
• This turns the expression into \((4 + (-4))(-2)\).

Simply replace each variable with the corresponding value each time it appears in the expression. It's like following a recipe with specific ingredients.

Simplifying Expressions

Once all variables are substituted, the next step is to simplify the expression. Simplifying means breaking down the expression to its simplest form by performing basic arithmetic operations.
In the expression \((4 + (-4))(-2)\), we first handle the operation inside the parentheses:

• Calculate \(4 + (-4)\), which equals \(0\).
• The expression now becomes \(0 \times (-2)\).

By simplifying, we make the expression easier to evaluate, reducing it to something manageable. Simplifying also helps ensure that all calculations are clear and correct.

The Multiplication Property of Zero

The multiplication property of zero is a fundamental rule in mathematics. It states that when any number is multiplied by zero, the result is always zero.
In our expression \(0 \times (-2)\), we see this very rule come into play:

• No matter what value a number has, if you multiply it by \(0\), the answer is \(0\).
• In this example, multiplying \(0\) and \(-2\) gives us \(0\).

This property helps make calculations simpler by quickly reducing parts of an expression to zero without further complications. It's a great shortcut when dealing with expressions where a factor ends up being zero.