Problem 54 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(|5 z-2 y|\)

Step-by-Step Solution

Answer

The value of the expression is 19.

Step 1: Substitute the Values

First, substitute the given values into the expression. For \(z = 5\) and \(y = 3\), the expression \(|5z - 2y|\) becomes \(|5(5) - 2(3)|\).

Step 2: Simplify Inside the Absolute Value

Calculate the expressions inside the absolute value. Compute \(5 \times 5 = 25\) and \(2 \times 3 = 6\). This changes the expression to \(|25 - 6|\).

Step 3: Calculate the Difference

Subtract the two numbers inside the absolute value: \(25 - 6 = 19\).

Step 4: Apply the Absolute Value

Apply the absolute value to the result of the subtraction. Since 19 is already positive, \(|19| = 19\).

Key Concepts

Evaluating Expressions

Evaluating expressions involves finding the value of a mathematical phrase. This phrase may include numbers, variables, and operations, such as addition or multiplication. In this context, you are looking at the expression \(|5z - 2y|\). Evaluation requires following specific steps to arrive at a final numerical value.

• Identify the variables in the expression. Here we have variables \(z\) and \(y\).
• Determine what operations will be performed. In this expression, there are multiplication and subtraction operations.
• Understand that the absolute value denotes the non-negative value of an expression, meaning how far it is from zero, regardless of direction on the number line.

Understanding these basics steps will help you correctly evaluate expressions, whether they are inside or outside of an absolute value.

Substitute Values

Substitution is essentially replacing the variables with given numbers. It's like swapping placeholders with their true value. In our expression, you will replace \(z=5\) and \(y=3\) into \(|5z - 2y|\):

• Replace \(z\) with 5, the value provided for \(z\).
• Replace \(y\) with 3, the value given for \(y\).

This substitution transforms the expression into \(|5(5) - 2(3)|\). Now, you have a numerical expression to work with instead of a variable-based one. This step is necessary to progress to further simplification.

Simplifying Expressions

Once substitution is complete, simplifying the expression is the next step. Simplification involves performing operations inside the expression to make it as straightforward as possible.

• First, deal with multiplication: Calculate \(5 \times 5 = 25\) and \(2 \times 3 = 6\).
• Next, perform subtraction within the absolute bars: Subtract \(6\) from \(25\), resulting in \(19\).

By following these steps, you change \(|5(5) - 2(3)|\) into \(|25 - 6|\), and then to \(|19|\). This makes the process of finding the final value more straightforward.

Integer Operations

Integer operations are fundamental arithmetic operations, using whole numbers, which include addition, subtraction, multiplication, and division. In this scenario, you are performing multiplication and subtraction.

• Multiply: Execute multiplication \((5 \times 5)\) and \((2 \times 3)\) to handle initial operations.
• Subtract: After multiplication, subtract the products \(25 - 6\) to simplify within the expression.

Finally, apply the absolute value operation, which turns any negative result positive. Since our result, \(19\), is already positive, we get \(|19| = 19\). This use of absolute value ensures a non-negative result, highlighting it on the number line.