Problem 35 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ \frac{m}{6}+5 m, \text { if } m=-18 $$

Step-by-Step Solution

Answer

The value of the expression is \(-93\).

Step 1: Substitute the Value of m

The expression given is \( \frac{m}{6} + 5m \). We need to substitute \( m = -18 \) into this expression. When we substitute, it becomes \( \frac{-18}{6} + 5(-18) \).

Step 2: Simplify the Fraction

Now, divide \(-18\) by \(6\) which results in \(-3\). So the expression becomes \(-3 + 5(-18) \).

Step 3: Simplify the Multiplication

Calculate \( 5 \times (-18)\), which is \(-90\). This gives us \(-3 - 90\).

Step 4: Combine the Results

Add \(-3\) and \(-90\) to get the final result: \(-93\).

Key Concepts

Substitution

Substitution in algebra is a crucial step to solving problems involving variables. The goal is to replace a variable with a given number to simplify the expression. In this exercise, you start with an expression that contains the variable \( m \): \( \frac{m}{6} + 5m \). You are told that \( m = -18 \), and the task is to find the value of the expression using this information. To substitute, replace every occurrence of \( m \) in the expression with \(-18\).

This process is fairly straightforward but essential for solving the problem. Once you substitute, you have a new expression without variables: \( \frac{-18}{6} + 5(-18) \). Now, you can move on to the next steps to simplify and solve this expression.

Simplification

Simplification is the process of making an expression easier to solve or understand. In this exercise, once you have substituted \( m \) with \(-18\), the expression takes the form \( -3 + 5(-18) \). Simplifying involves performing mathematical operations like addition, subtraction, multiplication, or division to make the expression concise.

Here, you start by simplifying the fraction \( \frac{-18}{6} \), which yields \(-3\). Removing the fraction simplifies the problem, allowing you to focus on the remaining terms to obtain \(-3 + 5(-18)\).

• Simplifying fractions and performing arithmetic operations are the keys to solving the problem correctly.
• The simplified expression leads directly to calculating the final result easily.

Be thorough in each step to avoid mistakes and get the correct answer.

Fractions

Fractions can make algebraic expressions seem daunting, but they often simplify down with a bit of arithmetic. In this example, you encounter the fraction \( \frac{-18}{6} \) as part of the algebraic expression. Dealing with fractions requires dividing the numerator by the denominator.

When you divide \(-18\) by \(6\), you simplify the fraction to \(-3\). This step is essential as it reduces the complexity of the expression.

• Simplifying fractions clarifies further arithmetic operations.
• Understand that dividing by a positive number flips the sign if the numerator is negative.

Once fractions are simplified, the problem becomes more about straightforward arithmetic, setting you up for solving the final step.

Multiplication

Multiplication in algebra involves combining terms based on their coefficients and signs. In this problem, after simplifying the fraction, we reach the multiplication part: \( 5(-18) \). At this stage, you simply multiply the constant \(5\) by \(-18\).

Remember the rule: multiplying a positive number by a negative number results in a negative product. Therefore, \( 5 \times (-18) = -90 \). After performing this multiplication, you'll have \(-3 - 90\) remaining, which is set for final calculation.

• Pays close attention to signs during multiplication to avoid errors.
• The negative result stems from the rule about signs in multiplication.

After these calculations, you can finally add the results and find the expression's total value.