Problem 84 - Algebra 2

Question

Evaluate each expression for the given values of the variable. \(4 m+7 ; m=1,2,3,4,5\)

Step-by-Step Solution

Answer

The expression evaluates to 11, 15, 19, 23, and 27 for \(m = 1, 2, 3, 4, 5\), respectively.

Step 1: Understand the Expression

The expression given is \(4m+7\). This means for each value of \(m\), we need to multiply 4 by \(m\) and then add 7 to the result.

Step 2: Evaluate for \(m = 1\)

Substitute \(m = 1\) into the expression: \(4(1) + 7\). Calculate: \(4 \times 1 = 4\), then \(4 + 7 = 11\). So, the expression evaluates to 11 when \(m = 1\).

Step 3: Evaluate for \(m = 2\)

Substitute \(m = 2\) into the expression: \(4(2) + 7\). Calculate: \(4 \times 2 = 8\), then \(8 + 7 = 15\). So, the expression evaluates to 15 when \(m = 2\).

Step 4: Evaluate for \(m = 3\)

Substitute \(m = 3\) into the expression: \(4(3) + 7\). Calculate: \(4 \times 3 = 12\), then \(12 + 7 = 19\). So, the expression evaluates to 19 when \(m = 3\).

Step 5: Evaluate for \(m = 4\)

Substitute \(m = 4\) into the expression: \(4(4) + 7\). Calculate: \(4 \times 4 = 16\), then \(16 + 7 = 23\). So, the expression evaluates to 23 when \(m = 4\).

Step 6: Evaluate for \(m = 5\)

Substitute \(m = 5\) into the expression: \(4(5) + 7\). Calculate: \(4 \times 5 = 20\), then \(20 + 7 = 27\). So, the expression evaluates to 27 when \(m = 5\).

Key Concepts

Evaluation of Expressions

Evaluating expressions is a fundamental skill in algebra. It means finding the value of an expression by replacing variables with given numbers. This step is like transforming letters into numbers to solve a puzzle. Imagine you have a formula, just like in math class. When evaluating, you substitute numbers in place of the letters or variables.
To evaluate an expression like \(4m + 7\), we perform these operations:

• Identify the expression and the value of the variable \(m\).
• Substitute the number for \(m\), and replace it wherever \(m\) appears in the expression.
• Carry out the arithmetic operations in the correct order, respecting the order of operations.

For instance, when \(m = 3\) in the expression \(4m+7\), replace \(m\) with 3, then calculate: \(4 \times 3 = 12\), followed by adding 7, to get 19.

Substitution in Algebra

Substitution is a strategy used in algebra to solve expressions by replacing variables with their assigned values. It is like swapping a letter for a number in an equation. This step allows you to evaluate an expression for different scenarios.

• Start by identifying what each variable represents.
• Insert the specific number in place of the variable in your expression.
• Recalculate the expression with numbers instead of variables.

Using substitution, an algebraic sentence becomes a numeric one.
In substituting variables, like placing \(m = 4\) into \(4m + 7\), you directly multiply wherever you see \(m\) with 4, \(4 \times 4 = 16\), then add 7 to get 23. This transforms the algebraic expression into a straightforward calculation.

Variables in Algebra

Variables are a basic building block in algebra. They are symbols, usually letters, that stand in for unknown or changeable numbers. Think of a variable as a placeholder waiting to be filled with different values.
When you work with variables:

• You use them to represent numbers in expressions and equations.
• They allow you to work with formulas that are flexible and applicable in various situations.
• Variables make it possible to describe a range of values with a single expression.

In the expression \(4m+7\), \(m\) is the variable. It can change and take different values, like 1, 2, 3, etc. Each time \(m\) changes, it offers a new perspective on the same algebraic rule, letting you calculate different outcomes as seen in evaluating expressions for various \(m\) values.