Problem 68 - Prealgebra

Question

What is the value of \(\frac{3(4 a-3 b)}{b-4}\) if \(a=6\) and \(b=7 ?\)

Step-by-Step Solution

Answer

The value is 3.

Step 1: Substitute the Values

Replace the variables in the expression \(\frac{3(4a-3b)}{b-4}\) with the given values \(a=6\) and \(b=7\).

Step 2: Simplify Inside the Parentheses

Calculate inside the parentheses: \(4a - 3b = 4(6) - 3(7) = 24 - 21 = 3\).

Step 3: Multiply

Multiply the result from Step 2 by 3: \(3 \times 3 = 9\).

Step 4: Compute the Denominator

Calculate the denominator: \(b-4 = 7-4 = 3\).

Step 5: Divide the Results

Divide the result from Step 3 by the result from Step 4: \(\frac{9}{3} = 3\).

Key Concepts

Substitution Simplified

Substitution is like playing a little game where we replace letters with numbers in an algebraic expression. Think of it as following simple instructions to find out what expressions are worth when certain conditions apply. In our exercise, we have an expression with letters: \( \frac{3(4a - 3b)}{b - 4} \). The game is to substitute \( a \) and \( b \) with 6 and 7 respectively.

• Instead of \( a \), we use the number 6.
• Instead of \( b \), we slot in the number 7.

Every single \( a \) magically turns into 6, and every \( b \) transforms into 7 when we substitute. This substitution trick helps us turn the expression into something concrete so we can do more math tricks later!

Simplifying Expressions with a Smile

After substituting numbers for variables, the next fun task is to simplify the expression. It's like tidying up your room—organizing everything so it looks nice and neat. In this case, after substitution, our expression changes to \( \frac{3(4(6) - 3(7))}{7 - 4} \). Let’s focus on simplifying the expression within the parentheses, \(4(6) - 3(7)\).

• Multiply the numbers with the parentheses: \(4 \times 6 = 24\) and \(3 \times 7 = 21\).
• Subtract the results: \(24 - 21 = 3\).

Simplifying the inside first makes the math easier to handle. Then, we just multiply the result by 3, giving us 9. See how following steps bring clarity and order?

Order of Operations: PEMDAS’s Golden Rule

In math, respecting the order of operations is key to solving expressions correctly. We follow a specific sequence, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

• First, handle operations inside Parentheses: 4(6) - 3(7).
• Next, perform the Multiplication, then Division: Multiply the result, 3, by 3, then divide by the number in the denominator.

Our expression, which we simplified to \(\frac{9}{3}\), gets solved by division, leading us to the final answer. Respecting PEMDAS prevents misunderstandings and helps avoid silly mistakes, making us math heroes every time!