Problem 79 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 13.

Step 1: Substitute Values

Start by substituting the given values for each variable in the expression. We have the expression \(-z + 3x\). Substitute \(x = 3\), \(y = -2\), and \(z = -4\) into the expression. This transforms the expression into \(-(-4) + 3 \times 3\).

Step 2: Simplify Negatives

Simplify the negative signs in the expression. Since we have \(-(-4)\), remember that two negatives make a positive, so change \(-(-4)\) to \(4\). Now the expression becomes \(4 + 3 \times 3\).

Step 3: Multiply

Perform the multiplication next. Multiply \(3\) by \(3\) to get \(9\). So now the expression simplifies to \(4 + 9\).

Step 4: Add

Finally, add the values together. Add \(4\) and \(9\) to get \(13\). Thus, the expression evaluates to \(13\).

Key Concepts

Substitution Method

The substitution method is a powerful technique used in algebra to evaluate expressions by replacing variables with their given numerical values. Imagine it as filling in the blanks, where each letter or symbol in the expression has a specific number designated to it. In this method, we closely follow these steps:

• Identify all the variables in the expression.
• Use the provided values to replace each variable. For instance, if you have an expression like \(-z + 3x\), and you know \(z = -4\) and \(x = 3\), substitute directly so the expression becomes \(-(-4) + 3 imes 3\).

This method turns an abstract expression into something you can easily compute. By using substitution, you simplify the process of evaluating complex algebraic expressions, making them more manageable and straightforward.

Simplifying Expressions

Simplifying expressions is all about making mathematical expressions easier to work with. It's like putting a puzzle together, where simplifying means assembling all the pieces to see the bigger picture.

• Start with handling negative signs. Remember that two negatives make a positive. For example, if you have \(-(-4)\), change it to \(4\).
• Combine like terms when possible. This means taking all similar pieces of the expression and adding them together to make computation easier.

Simplification is crucial because it reduces complexity, transforming the expression into a form where further operations, like multiplication or addition, are more easily performed.

Order of Operations

The order of operations is a fundamental concept in algebra that dictates the sequence in which various operations should be performed to accurately evaluate an expression. Think of it as a roadmap or guide that directs you at each step.

• Firstly, handle any calculations inside parentheses.
• Next, deal with exponents (powers and roots).
• Then go for multiplication and division, moving from left to right across the expression.
• Finally, perform addition and subtraction, also working from left to right.

For the expression \(4 + 3 imes 3\), we first multiply \(3 imes 3\) to get \(9\), and then add \(4 + 9\) to arrive at \(13\). Remembering and adhering to the order of operations ensures that you evaluate expressions correctly each time.