Problem 52 - Beginning and Intermediate Algebra

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5.\) \(6 y-8\)

Step-by-Step Solution

Answer

10

Step 1: Substitute the Given Values into the Expression

Start with the expression given: \(6y - 8\). The task requires us to evaluate this expression for specific values of the variables. Substitute \(y = 3\) into the expression: \(6(3) - 8\).

Step 2: Calculate the Multiplication

Multiply 6 by 3, which results in 18. The expression now looks like \(18 - 8\).

Step 3: Perform the Subtraction

Subtract 8 from 18 to simplify the expression further. \(18 - 8 = 10\).

Step 4: Present the Final Result

The value of the expression \(6y - 8\) when \(y = 3\) is 10.

Key Concepts

Substitution

Substitution is a fundamental process in algebra that involves replacing variables in an expression with their actual numerical values. In our exercise, we substitute the value of the variable \( y \) with \( 3 \) in the expression \( 6y - 8 \). By doing this, we transform the expression into a numerical one, which can be easily solved. When substituting:

• Identify the variables and their given values.
• Replace each variable in the expression with its corresponding numerical value.
• Make sure to use parentheses when substituting numbers directly next to coefficients or other variables to avoid confusion. For instance, write \(6(3)\) instead of \(63\).

This technique is highly useful, making it possible to convert abstract algebraic expressions into concrete numbers that are easier to work with.

Evaluation of Expressions

Evaluating an expression means simplifying it down to a single numerical value using arithmetic operations. After substituting the known variable values into the expression, our next goal is to determine its outcome by performing the necessary calculations.Follow these steps for evaluating:

• Substitute all variables with their actual values, as mentioned earlier in the substitution process.
• Use the order of operations (PEMDAS/BODMAS) which stands for Parentheses/Brackets, Exponents/Orders, Multiplication & Division (from left to right), and Addition & Subtraction (from left to right).
• Calculate keeping the sequence of operations. For example, in \(6 \times 3 - 8\), first multiply \(6\) by \(3\) to get \(18\), then subtract \(8\) to reach the final value.

Evaluating expressions correctly ensures that you derive the accurate numerical answer from an algebraic setup.

Arithmetic Operations

Arithmetic operations are the basic calculations we perform on numbers and expressions in math. They include addition, subtraction, multiplication, and division. In algebra, these operations often involve variables.Key Operations:

• Multiplication: Combine numbers to form a product. For example, calculating \(6 \times 3\) gives \(18\).
• Subtraction: Deduct a number from another. After multiplying in our exercise, what remains is \(18 - 8\), which simplifies further to \(10\).

Grasping these operations is crucial, as they form the foundation for evaluating more complex algebraic expressions and solving equations efficiently. With clear understanding and practice, arithmetic becomes an essential tool in every math problem you face.