Solving Algebraic Equations: Finding the Two Numbers That Add Up to 29 and Subtract to 5

Algebra can often challenge even the most experienced problem solvers, but with the right techniques, tricky number puzzles can be untangled with ease. This article walks you through the process of solving a linear system to find two distinct numbers that, when added, yield a sum of 29, and when subtracted, result in 5. We'll explore multiple methods and provide step-by-step solutions so that you can master these essential algebraic skills.

Understanding the Problem: A Linear System of Equations

We are given two conditions to satisfy:

Two numbers, when added, equal 29. The same two numbers, when subtracted, equal 5.

Let's denote these unknown numbers as x and y. The goal is to solve the following system of linear equations:

x y 29 x - y 5

Method 1: Solving by Elimination

The simplest method for solving this system is by elimination. First, add the two equations:

(x y ) (x - y ) ( 29 ) ( 5 )

This simplifies to:

2x 34

Dividing both sides by 2, we get:

x 17

Next, substitute x 17 into the first equation:
17 y 29

Subtract 17 from both sides to find y :

y 29 - 17 12

Thus, the numbers are 17 and 12.

Alternative Method: Substitution (Solving for y)

Alternatively, you can solve for one variable and substitute it into the other equation. From the second equation, solve for y:

y x - 5

Substitute y x - 5 into the first equation:

x ( x - 5 ) 29

This simplifies to:

2x - 5 29

Adding 5 to both sides, we get:

2x 34

Dividing both sides by 2, we get:

x 17

Substitute x 17 into the equation for y x - 5:

y 17 - 5 12

Therefore, the numbers are 17 and 12.

Conclusion

By understanding the problem, applying the method of elimination, and validating the result, we find the two numbers that satisfy both conditions are 17 and 12. Whether using the elimination or substitution method, the solution remains consistent. These techniques are fundamental to solving linear systems and provide valuable insights into algebraic problem-solving.

Understanding how to solve such problems can significantly enhance your ability to tackle more complex equations and mathematical challenges. Always remember that breaking down the problem into manageable steps and verifying your results are key to success in algebra.