If p * q = 2p + pq + q, find p  when ( p * 2) - (p * 1) = 40

Analysis

The problem defines a binary operation: $p * q = 2p + pq + q$. We are given the equation $(p * 2) - (p * 1) = 40$ and need to solve for $p$.

Concept

A binary operation is simply a rule that tells you how to combine two numbers. To solve this, we must evaluate the two expressions $(p * 2)$ and $(p * 1)$ separately using the provided formula, substitute them back into the equation, and solve for $p$.

Step-by-Step Solution

Step 1: Evaluate $(p * 2)$ Using the formula $p * q = 2p + pq + q$, substitute $q = 2$: $(p * 2) = 2p + p(2) + 2$ $(p * 2) = 2p + 2p + 2$ $(p * 2) = 4p + 2$

Step 2: Evaluate $(p * 1)$ Using the same formula, substitute $q = 1$: $(p * 1) = 2p + p(1) + 1$ $(p * 1) = 2p + p + 1$ $(p * 1) = 3p + 1$

Step 3: Substitute into the given equation The equation is $(p * 2) - (p * 1) = 40$. $(4p + 2) - (3p + 1) = 40$

Step 4: Solve for $p$ Remove the brackets (be careful with the negative sign): $4p + 2 - 3p - 1 = 40$ Group the like terms: $(4p - 3p) + (2 - 1) = 40$ $p + 1 = 40$ $p = 40 - 1$ $p = 39$

Conclusion

By substituting the values into the binary operation formula and simplifying the resulting linear equation, we find that $p = 39$.

Exam Tip: When dealing with binary operations involving brackets, always expand the expressions inside the brackets first before performing the subtraction or addition. Also, watch out for the negative sign when subtracting an expression—it must be distributed to every term inside the second bracket!