Solving a Simultaneous Equation Puzzle: A Math Challenge

Math competitions can be thrilling, especially when they involve solving a series of puzzles that test a student's logical reasoning and mathematical skills. A typical competition might present a scenario like the one involving the scores of three participants: Kelvin Teddy, and Jim. Given the total score, relationships between their individual scores, and logical deductions, let's walk through how to solve the equation in a real-world math challenge.

The Problem Statement

Three participants, Kelvin, Teddy, and Jim, scored a combined total of 83 points during a competition. Teddy scored 13 points less than Kelvin, and Teddy scored 3 times as many points as Jim. The goal is to determine the individual scores of each participant.

Defining Variables and Setting Up Equations

To solve the problem, we first define the variables:

K - the score of Kelvin T - the score of Teddy J - the score of Jim

The total score equation is given by:

K T J 83

From the problem statement, we also know:

Teddy's score in relation to Kelvin's score: ( T K - 13 ) Teddy's score in relation to Jim's score: ( T 3J )

Solving the Equations

Next, we substitute the second and third equations into the first one:

K (K - 13) J 83

This simplifies to:

2K J - 13 83

Add 13 to both sides:

2K J 96

We can now substitute ( T 3J ) into the second equation:

3J K - 13

Rearranging gives:

K 3J 13

Substitute ( K 3J 13 ) into ( 2K J 96 ):

2(3J 13) J 96

This expands to:

6J 26 J 96

Combining like terms:

7J 26 96

Subtract 26 from both sides:

7J 70

Divide by 7:

J 10

Now, we can find ( T ) and ( K ):

Using ( T 3J ):

T 3 × 10 30

Using ( K 3J 13 ):

K 3 × 10 13 30 13 43

Summary of Results

So, the scores are:

Kelvin (K) scored 43 points. Teddy (T) scored 30 points. Jim (J) scored 10 points.

This systematic approach to solving simultaneous equations demonstrates the power of algebraic manipulation in real-world problem-solving scenarios. Such skills are highly valued in various fields, including math competitions, engineering, and data science.