If * is an operation such that a * b = 3a - 4b, find the value of 3 * 4

Question

Prepp Team · Accepted Answer

Understanding the Mathematical Operation

The question defines a new mathematical operation denoted by *. This operation takes two numbers, say 'a' and 'b', and relates them using the rule $a * b = 3 a - 4 b$ .

We are asked to find the value of $3 * 4$ .

Evaluating the Expression 3 * 4

To find the value of $3 * 4$ using the given operation rule $a * b = 3 a - 4 b$ , we need to substitute the values.

In the expression $3 * 4$ , we have:

The first number 'a' is 3.
The second number 'b' is 4.

Now, we substitute $a = 3$ and $b = 4$ into the rule $3 a - 4 b$ :

3 * 4 = 3 \times (3) - 4 \times (4)

Let's perform the multiplications:

3 \times 3 = 9

4 \times 4 = 16

Now substitute these results back into the expression:

3 * 4 = 9 - 16

Finally, perform the subtraction:

9 - 16 = - 7

So, the value of $3 * 4$ is -7.

Step-by-Step Calculation

Identify the given operation: $a * b = 3 a - 4 b$ .
Identify the values for 'a' and 'b' in the expression $3 * 4$ : $a = 3$ , $b = 4$ .
Substitute these values into the definition of the operation: $3 * 4 = 3 \times 3 - 4 \times 4$ .
Calculate the products: $3 \times 3 = 9$ and $4 \times 4 = 16$ .
Substitute the products back: $3 * 4 = 9 - 16$ .
Perform the final subtraction: $9 - 16 = - 7$ .

The result is -7.

Step	Operation	Calculation	Result
1	Define * operation	$a * b$	$3 a - 4 b$
2	Identify a and b for 3 * 4	a=3, b=4
3	Substitute values into definition	$3 \times 3 - 4 \times 4$
4	Perform multiplication	$9 - 16$
5	Perform subtraction		$- 7$

Revision Table: Custom Mathematical Operations

Concept	Description	Example
Custom Operation	A new rule defined for combining two or more numbers or elements, often represented by a symbol like *, #, @, etc.	$a * b = 3 a - 4 b$
Evaluating an Expression	Substituting given values into the definition of the operation and performing the required calculations following the order of operations.	Evaluating $3 * 4$ means calculating $3 \times 3 - 4 \times 4$ .
Order of Operations	Rules that define the sequence in which operations should be performed (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In this case, multiplication is done before subtraction.	Calculate $3 \times 3$ and $4\times4$ before $9 - 16$ .

Additional Information: Properties of Operations

Standard operations like addition (+) and multiplication ( $\times$ ) have properties such as commutativity ( $a + b = b + a$ ) and associativity ( $(a + b) + c = a + (b + c)$ ). Custom operations like the one defined might or might not have these properties.

Commutativity: Is $a * b = b * a$ ?
Using the definition: $a * b = 3 a - 4 b$ .
For $b * a$ , swap 'a' and 'b' in the definition: $b * a = 3 b - 4 a$ .
Since $3 a - 4 b$ is not generally equal to $3 b - 4 a$ (for example, we found $3 * 4 = - 7$ , but $4 * 3 = 3 (4) - 4 (3) = 12 - 12 = 0$ ), this operation is not commutative.
Associativity: Is $(a * b) * c = a * (b * c)$ ?
This would involve more calculations, but generally, custom operations do not possess standard properties unless specifically designed to do so.