Problem 75 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{7}{2} y-3 \text { when } y=4$$

Step-by-Step Solution

Answer

The result of evaluating the expression is 11.

Step 1: Substitute the variable

Replace the variable y in the expression with the given value, which is 4. You get: \((\frac{7}{2} * 4) - 3\)

Step 2: Perform the multiplication

According to PEMDAS/BODMAS, multiplication should be carried out before the subtraction operation. So, multiply \(\frac{7}{2}\) and 4 to get 14. You now have: \(14 - 3\)

Step 3: Perform the subtraction

Subtract 3 from 14 to simplify the expression. The result is 11.

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra, crucial for evaluating expressions with one or more variables. It involves replacing variables in an algebraic expression with their corresponding values. For instance, if you have the expression \(\frac{7}{2} y - 3\) and the variable \(y\) is given to be 4, you would substitute \(4\) into the expression in place of \(y\).

The process looks like this: \[\left(\frac{7}{2} \times 4\right) - 3\] After substitution, you continue with simplifying the expression following the correct order of operations, which leads to accurately solving the problem. The substitution method not only makes the evaluation process straightforward but also minimizes mistakes by providing a clear picture of the given situation. Ensuring correct and consistent substitution is essential, as any error in this step can lead to an incorrect answer in the end.

Order of Operations

Understanding the order of operations is integral to simplifying algebraic expressions correctly. The standard rule, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. This sequence ensures that everyone simplifies expressions the same way and gets the same results.

In evaluating the expression \(\frac{7}{2} y - 3\), after substituting \(y\) with 4, the next step is to address the multiplication before the subtraction: \[\left(\frac{7}{2} \times 4\right) - 3 = 14 - 3\] Once you've completed the multiplication, you proceed with the subtraction. The clarity the order of operations brings to the table is invaluable, especially as expressions become more complex.

Simplifying Expressions

Simplifying expressions involves performing all the possible calculations within an expression to write it in its simplest form. This doesn't always mean obtaining a numerical answer; for some expressions, simplifying could mean factoring or combining like terms to make them more manageable.

After substitution and following the order of operations, the numerical expression \(14 - 3\) is then simplified to get the final answer. The simplification here involves straightforward arithmetic; subtracting 3 from 14 gives you 11. In the context of algebra, simplification is powerful—it reveals the most reduced form of the expression and is essential to solution clarity and correctness.