Question

For all numbers a and b, the operation ? is defined by a ? b = (a + 2)(b ? 3). If 3 ? x = ?45, then x =

Short answer

x = -6.

Step-by-step solution

Understand the Operation

The operation ? is defined such that for any numbers 'a' and 'b', the formula is: a ? b = (a + 2)(b ? 3).This defines how to perform the custom operation on two numbers.

Set Up the Given Equation

Given that 3 ? x = ?45, substitute 3 for 'a' and 'x' for 'b' in the operation’s definition.3 ? x = (3 + 2)(x ? 3) = 5(x ? 3).

Solve for x

Equate the expression to ?45 and solve for x:5(x ? 3) = ?45.Divide both sides by 5:x ? 3 = -9.Add 3 to both sides:x = -6.

Key concepts

algebraic equations

Algebraic equations are mathematical statements that show the relationship between two expressions with an equal sign.
They often involve variables, which are symbols that represent unknown values.
In our example, we are dealing with an equation involving a custom operation. This operation is defined for any numbers 'a' and 'b' as follows: \(a ? b = (a + 2)(b ? 3)\).
Realize that the goal is to find the value of 'x' in the equation 3 ? x = -45. By substituting 'a' with 3 and 'b' with 'x', the operation simplifies to 5(x - 3).
Hence, we form the equation 5(x - 3) = -45.

problem-solving steps

To solve any mathematical problem systematically, follow these essential steps:
Identify and understand the operation or equation.
Set up the equation with given values or variables.
Solve for the unknown variable.
In our problem of 3 ? x = -45, we:

• First understand the operation, defined as \(a ? b = (a + 2)(b ? 3)\).
• Substitute the given values into the operation: (3 + 2)(x ? 3) = 5(x ? 3).
• Form the equation 5(x ? 3) = -45 to solve for 'x'.

To isolate 'x', divide both sides by 5 to get x ? 3 = -9.
Finally, add 3 to both sides to find x, so x = -6.

mathematical definitions

In mathematics, it is crucial to understand definitions to solve problems efficiently:

• Operations: Actions performed on numbers or variables defined by specific rules. In our case, the custom operation \(a ? b = (a + 2)(b ? 3)\) is such an action.
• Equation: A statement that two expressions are equal. For example, 3 ? x = -45 forms an equation we solve for x.
• Variable: A symbol representing an unknown value, like 'x' in our exercise.

Understanding these basic terms lays the foundation for approaching and solving algebraic problems.
By grasping these concepts, we can navigate through any custom operation or algebraic equation confidently.