Problem 4 - Algebra and Trigonometry

Question

Evaluate each algebraic expression for the given value or values of the variable(s). $$8 x-y,\quad for\quad x=3\quad and\quad y=4$$

Step-by-Step Solution

Answer

The result of evaluating the algebraic expression \(8x - y\) for \(x=3\) and \(y=4\) is 20

Step 1: Substitute the given values of variables into the equation

Substitute \(x = 3\) and \(y = 4\) into the algebraic expression \(8x - y\). This gives us \(8*3 - 4\), which simplifies to \(24 - 4\).

Step 2: Complete the subtraction operation

The operation \(24 - 4\) results in 20. This is the answer we are looking for.

Step 3: Validate the answer

This step involves validating the answer from the previous step. Since there are no other variables in the equation, and we have substituted all given values correctly, our answer should be correct.

Key Concepts

Understanding Variable Substitution

Variable substitution is a fundamental concept in algebra, involving the replacement of variables with actual numerical values. This process often simplifies expressions and allows for the calculation of definite results.
First, identify the variables in the given expression. In our exercise, we have the expression \(8x - y\) with variables \(x\) and \(y\).

• Assign a specific number to each variable as specified. Here, \(x = 3\) and \(y = 4\).
• Substitute these values back into the original expression. This means wherever there is an \(x\), we replace it with \(3\), and wherever there is a \(y\), substitute with \(4\).

After substitution, the expression \(8x - y\) becomes \(8 \times 3 - 4\). This step makes the previously abstract expression understandable in concrete terms, paving the way for further simplification and solving.

The Process of Simplification

Simplification is all about making expressions as straightforward and comprehensible as possible. Once you've substituted variables with their numerical values, the next step is breaking down the expression.
For the expression \(8 \times 3 - 4\):

• Start by performing any multiplication or division first, following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• In our case, multiply \(8\) by \(3\), resulting in \(24\).
• After completing the multiplication, proceed with subtraction: \(24 - 4\).

This careful method leads to an expression's simplest form, facilitating easier solution finding and error checking.

Basic Arithmetic Operations in Algebraic Expressions

Arithmetic operations are the building blocks of algebraic expressions and include addition, subtraction, multiplication, and division. When working with algebraic expressions like \(8x - y\), these operations help in deriving a numerical outcome.

• Multiplication: This was used first after substituting the values, multiplying \(8\) by the value of \(x\), which is \(3\).
• Subtraction: Following the multiplication, we performed subtraction, deducting \(4\) from \(24\).

These operations require careful attention, especially when dealing with multiple terms or operations together. Consistently practicing these basics through exercises refines your skills and boosts confidence in handling algebraic expressions.