Problem 58 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. $$ 7 \cdot y \text { when } y=8 $$

Step-by-Step Solution

Answer

The evaluated expression \(7 \cdot y\) when \(y=8\) is 56.

Step 1: Understand the Given Variable

In this expression, we are given that \(y = 8\).

Step 2: Substitute the Value of the Variable

Substitute \(y = 8\) into the expression. This gives us: \(7 \cdot y = 7 \cdot 8\).

Step 3: Perform the Operation

Now, simply perform the multiplication operation: \(7 \cdot 8 = 56\).

Key Concepts

Variable Substitution

When dealing with algebraic expressions, one of the first steps you often need to take is variable substitution. This means replacing a variable in an expression with a specific value that has been provided.
For example, if you have the expression \(7 \cdot y\) and you're asked to find its value when \(y = 8\), you're going to substitute 8 in place of the variable \(y\). So the expression changes from \(7 \cdot y\) to \(7 \cdot 8\).
This process is crucial because it allows you to evaluate the expression with actual numbers instead of just variables, giving you a concrete numerical answer.

• Identify the variable and its given value.
• Carefully replace the variable in the expression with the given value.

Multiplication

After substituting the variable with its given value, your next step is usually multiplication, especially in expressions that involve the multiplication of numbers and variables.
Once you've substituted the value into the expression, like turning \(7 \cdot y\) into \(7 \cdot 8\), you then perform the multiplication to simplify the expression to a single number.
Multiplication within algebra helps in combining numbers efficiently:

• If you have \(7 \cdot 8\), you will multiply 7 and 8 using standard multiplication.
• Start by multiplying each number, paying careful attention to the basic multiplication table.
• The product, \(56\) in this example, represents the value of the original expression given the specified variable value.

Evaluation of Expressions

The final step in solving the exercise is the evaluation of expressions. This refers to the entire process of finding the value of an algebraic expression once the variable has been substituted and necessary operations, like multiplication, have been performed.
In our problem, the evaluation process involved the substitution of \(y\) with 8 and then multiplying, leading to the final result of \(56\).
This step is crucial because it checks whether or not the operations have been executed correctly.

• Ensure all values are correctly substituted.
• Verify that all operations, such as multiplication, are accurately completed.
• The goal is to simplify the expression into one final, simplified numerical answer.