Problem 39 - Fundamentals of Mathematics

Question

For problems \(17-46\), find the value of each expression. $$ \frac{-6 a}{5}+3 a+10, \text { if } a=25 $$

Step-by-Step Solution

Answer

The value of the expression is 55.

Step 1: Substitute the Given Value

In the expression \( \frac{-6a}{5} + 3a + 10 \), replace \( a \) with 25, as specified in the problem. This gives us \( \frac{-6(25)}{5} + 3(25) + 10 \).

Step 2: Simplify Each Term

Calculate the value of each term separately: 1. \( \frac{-6(25)}{5} = \frac{-150}{5} = -30 \).2. \( 3(25) = 75 \).3. The constant term is \( 10 \).

Step 3: Combine all terms

Add the three values obtained in step 2: \(-30 + 75 + 10\).

Step 4: Calculate the Final Result

Add the calculated values: \(-30 + 75 = 45\) and \(45 + 10 = 55\).

Key Concepts

Substitution Method

The substitution method is a technique used in algebra to simplify expressions and solve equations. The main idea is to replace variables with given values, making it easier to perform calculations.

• Identify the variable to be replaced. In our problem, it is "a".
• Substitute the given value into the expression. Here, replace "a" with 25.
• Make sure to replace the variable consistently throughout the expression.

By substituting the value of "a" into the expression \( \frac{-6a}{5} + 3a + 10 \), we transform it into a numeric expression: \( \frac{-6(25)}{5} + 3(25) + 10 \). This approach helps simplify the calculation by turning it into simple arithmetic.

Simplifying Expressions

Simplifying expressions is about breaking down complex mathematical phrases into simpler forms. This is necessary to make calculations easier, clearer, and quicker.

• Handle one operation at a time. Start with multiplication and division for terms that include them.
• Convert each part to its simplest form before moving on to addition and subtraction.

For our expression: - Calculate \( \frac{-6(25)}{5} \) first. Multiply \(-6\) by 25 to get \(-150\) and then divide by 5 to simplify to \(-30\).
- Next, multiply \(3(25)\) to get 75. - The constant term, 10, remains the same as it is already simplified.
This step-by-step simplification sets the stage for effortless arithmetic operations to find the final result.

Arithmetic Operations

Arithmetic operations are the building blocks of algebra. They include basic calculations like addition, subtraction, multiplication, and division. Here's how to handle them:

• Start with operations already performed in simplified expressions, like multiplying and dividing first.
• Proceed with addition and subtraction in the order of appearance.

In the simplified expression, \(-30 + 75 + 10\), we apply these operations: - Add the negative and positive values together: \(-30 + 75\) resulting in 45.
- Finally, add 10 to 45, reaching the result of 55.
These operations follow the left-to-right rule in the order of operations, also known as PEMDAS/BODMAS, ensuring calculations are accurate and ordered logically.