Some equations hide a system of relationships among several unknowns, which may seem more difficult than they actually are. The main challenge lies in properly organizing the information to find the values of each variable before using them in a final expression. This type of exercise requires methodical thinking, precision, and good management of calculation steps.
As Isaac Newton once said:
“Truth always resides in the simplicity of principles.”
The goal is not merely to calculate, but to understand how two equations can complement each other to determine all unknowns, then apply these results in a final expression while respecting the order of operations.
Before diving into the heart of the matter, let’s see what this challenge allows us to work on:
- The resolution of systems of equations
- The isolation of unknowns
- Rigor in intermediate calculations
- Respect for the order of operations
- Verification of the final result
Today’s Challenge
Answer: ______ ??
Take your time to carefully analyze the system before responding.
A good strategy is to first determine the values of a and b, then substitute them into the final equation.
The countdown begins:
40…
30…
20…
10…
5…
3…
2…
1…
Time’s up!
So, what is your answer?
What Your Performance Reveals
If you found the result quickly and correctly, you possess excellent logic and analytical skills.
If you arrived at the result after some thought, you know how to structure your reasoning and verify your steps.
And if your result was incorrect or not found, no problem; this type of exercise is perfect for improving your method and precision.
Detailed Solution We Found (Not Necessarily the Only One, Sometimes Even the Correct One).
1) Solve the System
We have:
- a + b = 26
- b − a = 12
We add the two equations:
(a + b) + (b − a) = 26 + 12
2b = 38
b = 19
We replace in a + b = 26:
a + 19 = 26
a = 7
2) Calculate the Equation
6a − 3b × 2
We respect the order of operations: multiplication first
- 3b × 2 = 6b
Thus:
6a − 6b
We substitute:
- 6 × 7 = 42
- 6 × 19 = 114
42 − 114 = -72
Final Answer:
-72
