Problem 82 - Algebra 2

Question

Evaluate each expression for the given values of the variable. \(3 n-1 ; n=1,2,3,4\)

Step-by-Step Solution

Answer

The values are 2, 5, 8, and 11 for \(n = 1, 2, 3, 4\) respectively.

Step 1: Understanding the Expression

The expression given is \(3n - 1\), which tells us to multiply the variable \(n\) by 3 and then subtract 1. Next, we are asked to evaluate this expression for the values \(n=1\), 2, 3, and 4.

Step 2: Substituting for \(n = 1\)

Substitute \(n = 1\) into the expression: \(3 \times 1 - 1\). Calculate the result of this substitution step-by-step.

Step 3: Calculating for \(n = 1\)

Perform the multiplication: \(3 \times 1 = 3\). Then subtract 1: \(3 - 1 = 2\). So, when \(n = 1\), the expression evaluates to 2.

Step 4: Substituting for \(n = 2\)

Substitute \(n = 2\) into the expression: \(3 \times 2 - 1\). Proceed with the calculation.

Step 5: Calculating for \(n = 2\)

Perform the multiplication: \(3 \times 2 = 6\). Then subtract 1: \(6 - 1 = 5\). So, when \(n = 2\), the expression evaluates to 5.

Step 6: Substituting for \(n = 3\)

Substitute \(n = 3\) into the expression: \(3 \times 3 - 1\). Then calculate the final result.

Step 7: Calculating for \(n = 3\)

Perform the multiplication: \(3 \times 3 = 9\). Then subtract 1: \(9 - 1 = 8\). So, when \(n = 3\), the expression evaluates to 8.

Step 8: Substituting for \(n = 4\)

Substitute \(n = 4\) into the expression: \(3 \times 4 - 1\). Finally, compute the result.

Step 9: Calculating for \(n = 4\)

Perform the multiplication: \(3 \times 4 = 12\). Then subtract 1: \(12 - 1 = 11\). So, when \(n = 4\), the expression evaluates to 11.

Key Concepts

Substitution Method

The substitution method is an essential tool in algebra. It involves replacing a variable with a specific value. In our exercise, we have the expression \(3n - 1\), and we need to find out what it equals for various values of \(n\).

• Start by identifying the variable in your expression, which, in this case, is \(n\).
• Substitute each given value of \(n\) into the expression one at a time.
• For example, for \(n=1\), you'd substitute 1 wherever you see \(n\), changing the expression to \(3 \times 1 - 1\).

This process requires patience as you work through each value, ensuring you replace the variable correctly each time. It's a powerful method when dealing with multiple values.

Multiplication

Multiplication is one of the core operations in this exercise. Here, it's applied to the expression \(3n - 1\). The number 3 is multiplied by the value of the variable \(n\).

• Understand that multiplication is about scaling or repeating groups. So, \(3 \times n\) means you're adding \(n\) three times.
• When substituting \(n\) with a number, your steps would first include performing the multiplication part: for instance, \(3 \times 2 = 6\) if \(n = 2\).

Mastering multiplication with variables is key because it pops up frequently in algebraic problems. It’s important to perform it accurately before proceeding with any other operations.

Subtraction

In our expression \(3n - 1\), subtraction comes into play after completing the multiplication step. It's essential to perform this step to get the correct result for the expression.

• Once you've multiplied, you take that product and subtract 1 from it, which is part of simplifying the expression.
• For \(n=3\), the process would be: after calculating \(3 \times 3 = 9\), subtract 1 to get \(9 - 1 = 8\).

Subtraction helps us decrease the value obtained from multiplication, and it follows a left-to-right sequence in operations, affecting the final outcome.

Variable Evaluation

Variable evaluation involves determining the value of an expression by substituting variables with actual numbers. In evaluating expressions like \(3n - 1\), this is the final goal.

• A key step is substituting the variable, as already described, leading directly to calculating the result.
• By substituting and then performing operations like multiplication and subtraction, you can find the expression's value for a specific \(n\).
• For instance, by evaluating with \(n=4\), your expression becomes \(3 \times 4 - 1\), eventually simplifying to 11.

This evaluation allows you to see how the expression changes with different variable values, offering insights into the relationship between variables and their real-number counterparts.