Problem 40 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ \frac{-7 h}{9}-7 h-7, \text { if } h=-18 $$

Step-by-Step Solution

Answer

The value of the expression is 133.

Step 1: Substitute the Value of h

Start by substituting the given value of \( h = -18 \) into the expression \( \frac{-7h}{9} - 7h - 7 \). This gives: $$ \frac{-7(-18)}{9} - 7(-18) - 7. $$

Step 2: Simplify Each Term

Calculate each term separately. First, simplify \( \frac{-7(-18)}{9} \):\[ \frac{-7(-18)}{9} = \frac{126}{9} = 14. \]Then, calculate \(-7(-18)\):\[ -7(-18) = 126. \]The last term is simply \(-7.\)

Step 3: Add Together the Simplified Terms

Combine the results from Step 2:\[ 14 + 126 - 7. \]

Step 4: Final Calculation

Compute the expression obtained in Step 3:\[ 14 + 126 - 7 = 133. \]

Key Concepts

Substitution

In the world of algebra, substitution is a common technique used to simplify complex expressions. In this exercise, we're provided with the expression \( \frac{-7h}{9} - 7h - 7 \) and a value for \( h \). By substituting \( h = -18 \) into the equation, we replace every occurrence of \( h \) with \(-18\). This process allows us to transform a general expression into one that is specific and solvable. Substitution involves:

• Identifying the variable to replace.
• Replacing the variable with its given numerical value.
• Rewriting the expression with these new values.

For our problem, substitution turns the expression into \( \frac{-7(-18)}{9} - 7(-18) - 7 \), setting the stage for simplification.

Simplification

Simplification is the process of making an expression easier to read and solve. After substitution, each part of the expression needs to be simplified. This means performing arithmetic operations to condense terms into simpler or singular values.In our exercise, we begin with the terms \( \frac{-7(-18)}{9} \) and \(-7(-18)\):

• The fraction \( \frac{-7(-18)}{9} \) simplifies to \( \frac{126}{9} \), which equals \( 14 \) after division.
• The product \(-7(-18)\) calculates to \( 126 \), changing a multiplication operation into a single value.
• The term \(-7\) stands alone, as it does not involve the variable \( h \).

These steps are crucial because they reduce the expression to numbers that are easily manipulated in further calculations.

Arithmetic Operations

Once the expression is fully simplified, arithmetic operations provide the final solution. Arithmetic operations include addition, subtraction, multiplication, and division—basic math actions that ensure we reach a final numeric result.In this final step, we combine all the simplified terms from the previous section:

• Add \( 14 \), \( 126 \), and subtract \( 7 \): this means calculating \( 14 + 126 - 7 \).
• The addition performs first, giving us \( 140 \).
• After subtraction, the final result is \( 133 \).

The efficiency of arithmetic operations is a cornerstone of algebra and makes it possible to solve expressions accurately and systematically.