What Are Examples Of System Of Linear Equations By Elimination Problems?

2025-07-20 10:42:14 356

3 Answers

Isaac
Isaac
2025-07-21 12:58:55
Solving linear equations by elimination reminds me of balancing scales—each step must maintain equality. A straightforward problem is x + y = 5 and x - y = 1. Adding these eliminates y, giving 2x = 6, so x = 3 and y = 2. This simplicity makes it great for beginners. For a twist, try 2x + 3y = 12 and 5x - 3y = 9. Adding them cancels y, leaving 7x = 21, so x = 3 and y = 2. The elegance lies in how variables vanish with strategic operations.

A slightly trickier system is 4x + 6y = 20 and 2x + 3y = 10. Scaling the second equation by 2 shows it’s Identical to the first, hinting at infinite solutions. Conversely, 3x + 2y = 7 and 6x + 4y = 15 would lead to a contradiction (0 = 1 after elimination), proving no solution exists. These examples showcase elimination’s versatility in diagnosing system behavior.
Uma
Uma
2025-07-23 01:40:38
Linear equations by elimination are a cornerstone of algebra, and I love how they mirror real-world problem-solving. Take the system 3x + 4y = 10 and 2x - y = 1. To eliminate y, multiply the second equation by 4, resulting in 8x - 4y = 4. Adding this to the first equation gives 11x = 14, so x ≈ 1.27. Plugging x back into the second equation yields y ≈ 1.54. This method is powerful because it systematically breaks down interconnected variables.

Another interesting case is 7x + 5y = 23 and 3x + 5y = 17. Subtracting the second equation from the first eliminates y, giving 4x = 6, so x = 1.5. Substituting x into either equation reveals y = 2.5. These problems highlight how elimination can untangle even tightly coupled variables. For a more challenging example, consider 6x + 9y = 15 and 4x + 6y = 10. Here, scaling and subtraction show the equations are dependent, meaning infinite solutions exist—a neat demonstration of how elimination reveals deeper system properties.
Penelope
Penelope
2025-07-25 19:54:37
I've always found elimination problems in linear equations fascinating because they feel like solving a puzzle. One classic example is a system like 2x + 3y = 8 and 4x - y = 6. To eliminate one variable, you can multiply the second equation by 3 to align the coefficients of y. This gives 12x - 3y = 18. Adding this to the first equation cancels out y, leaving 14x = 26, which simplifies to x ≈ 1.857. Substituting back gives y ≈ 1.429. Another problem could be 5x + 2y = 16 and 3x - 2y = 0. Here, adding the equations directly eliminates y, yielding 8x = 16, so x = 2 and y = 3. These examples show how elimination simplifies complex relationships into manageable steps.
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