Problem 22 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{9}{10} \cdot y-\frac{3}{10} \text { when } y=\frac{1}{2}$$

Step-by-Step Solution

Answer

The evaluated result is \(\frac{3}{20}\).

Step 1: Understand the Given Expression

The given expression is \(\frac{9}{10} \cdot y -\frac{3}{10}\). Here, the multiplication (\(\frac{9}{10} \cdot y\)) is done first followed by the subtraction (\(-\frac{3}{10}\)).

Step 2: Substitute the Given Value

The given value of the variable y is \(\frac{1}{2}\). Substitute this value into the given expression. Thus, the expression becomes \(\frac{9}{10} \cdot \frac{1}{2} -\frac{3}{10}\).

Step 3: Simplify the Expression

The multiplication should be done first as per the order of operations. Multiply \(\frac{9}{10}\) by \(\frac{1}{2}\) equals \(\frac{9}{20}\). Now subtract \(\frac{3}{10}\) from \(\frac{9}{20}\), which equals \(\frac{9}{20} - \frac{6}{20} = \frac{3}{20}\).

Key Concepts

Order of Operations

When evaluating algebraic expressions, it's crucial to follow the order of operations, a set of rules that dictate the sequence in which the operations should be performed. This is often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In the given exercise, we see a multiplication \(\frac{9}{10} \cdot y\) and a subtraction \( -\frac{3}{10}\). According to PEMDAS, we perform the multiplication before the subtraction. Knowing the order of operations helps prevent errors and ensures that everyone solves the expressions in the same way, arriving at the same result. Failing to follow this order could lead to incorrect answers and confusion.

Substituting Values

The process of substituting a given value into an algebraic expression is a fundamental skill in algebra. When you come across a variable in an expression, you replace it with the given numerical value before performing any operations. In our example, the variable \(y\) has been assigned the value \(\frac{1}{2}\).

It's important to substitute accurately and to ensure that the substituted value is placed correctly within the expression. The substitution in our problem turns \(\frac{9}{10} \cdot y\) into \(\frac{9}{10} \cdot \frac{1}{2}\), making the expression ready for the next step - simplification. Proper substitution is a simple yet vital step in evaluating expressions, and without it, the resulting answer will not reflect the true state of the solved expression.

Simplifying Expressions

Once the values have been substituted, the next step is simplifying the expression. Simplifying expressions means performing the operations and reducing the expression to its simplest form. This includes combining like terms, reducing fractions, and carrying out arithmetic operations.

In our exercise, after substituting \(y = \frac{1}{2}\) into the expression, we are left to multiply and subtract the fractions. We multiply \(\frac{9}{10} by \frac{1}{2}\) to get \(\frac{9}{20}\), and then we subtract \(\frac{3}{10}\), which is simplified by finding a common denominator and results in \(\frac{9}{20} - \frac{6}{20} = \frac{3}{20}\). Understanding how to simplify expressions ensures that you can efficiently reduce complex or lengthy expressions to their simplest forms, ultimately solving the problem at hand.