Problem 58 - Beginning and Intermediate Algebra

Question

Evaluate each expression if \(x=12, y=8,\) and \(z=4.\) \(\frac{y}{z}+8 x\)

Step-by-Step Solution

Answer

The expression evaluates to 98.

Step 1: Substitute Variable Values

First, substitute the given values of the variables into the expression. We have \(x=12\), \(y=8\), and \(z=4\). The expression becomes: \(\frac{8}{4} + 8 \times 12\).

Step 2: Evaluate the Division

Next, perform the division operation. Compute \(\frac{8}{4}\), which equals \(2\). Now the expression simplifies to: \(2 + 8 \times 12\).

Step 3: Evaluate the Multiplication

Multiply \(8\) by \(12\) to get \(96\). Replace the multiplication in the expression: \(2 + 96\).

Step 4: Perform the Addition

Finally, add \(2\) and \(96\) together to get \(98\). Thus, the value of the expression is \(98\).

Key Concepts

Substitution Method

The substitution method is a key technique in algebra that allows us to replace variables in an equation or expression with specific values. This is particularly useful when you're given specific values for variables, just like in our exercise. The process involves inserting the value of each variable directly into the expression.
This method helps simplify the algebraic expressions which can then be solved much like arithmetic problems.
In our given example:

• We start by substituting the values: for the expression \(\frac{y}{z}+8x\), with \(x=12\), \(y=8\), and \(z=4\).
• We replace each variable with its corresponding value which converts the expression into \(\frac{8}{4} + 8 \times 12\).
• This substitution reduces the complexity down to basic arithmetic operations, which are easier to handle.

To master this method, it's crucial to keep track of each variable and its value, ensuring that you're substituting accurately.

Evaluating Expressions

Evaluating expressions in algebra involves calculating the value of an expression after substituting the given variables. Once the substitution is done, we aim to simplify the expression to find its numerical value.
For this, follow the process carefully:

• First, substitute each variable with its value.
• Follow the necessary arithmetic operations such as division, multiplication, addition, or subtraction.

In our specific example, after substituting, we start by performing the division \(\frac{8}{4}\), which simplifies to \(2\). Then, you multiply \(8\) by \(12\) to get \(96\).
Finally, for the complete evaluation, add the results of these calculations: \(2 + 96\) which equals \(98\).
The goal of evaluating expressions is to simplify them step-by-step to reach a final numerical result.

Order of Operations

The order of operations is a fundamental principle in mathematics that dictates the correct sequence to solve expressions. This order is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
Applying this rule ensures that operations are performed in the proper order, without which we might get incorrect answers.
In our exercise, after substituting the given values, follow these steps:

• First, handle the division \(\frac{8}{4}\) to get \(2\).
• Next, perform the multiplication \(8 \times 12\) resulting in \(96\).
• Finally, apply addition to combine \(2\) and \(96\), giving \(98\).

Always remember to deal with any parentheses first, then exponents, followed by multiplication and division from left to right, and finally, address addition and subtraction. Following the order of operations prevents mistakes and assures a correct solution.