How To Check Solutions In System Of Linear Equations By Elimination?

Elimination is one of my favorite methods for solving systems of linear equations because it feels like a game of strategy. The key is to manipulate the equations so that adding or subtracting them eliminates one variable. Let's say you have the system:

3x + 2y = 8
x - y = 1

First, you might notice that if you multiply the second equation by 2, the y terms will have opposite coefficients. So, the second equation becomes 2x - 2y = 2. Now, add this to the first equation: 3x + 2y + 2x - 2y = 8 + 2. The y terms cancel, leaving 5x = 10, so x = 2. Substitute x back into the second original equation to find y: 2 - y = 1, so y = 1.

Sometimes, you need to multiply both equations to align the coefficients. For example, with 4x + 5y = 14 and 3x + 2y = 5, you could multiply the first by 3 and the second by 4 to make the x coefficients 12. Then subtract one from the other to eliminate x. The process is methodical, and checking your solution by plugging the values back into both equations ensures accuracy.

I remember learning this method in class, and it's actually pretty straightforward once you get the hang of it. The elimination method is about getting rid of one variable so you can solve for the other. You start by writing both equations clearly. Then, you adjust them so one of the variables cancels out when you add or subtract the equations. For example, if you have 2x + 3y = 5 and 4x + 6y = 10, you can multiply the first equation by 2 to match the coefficients of x. Then subtract the first from the second, and the x terms cancel out, leaving you with an equation in y. Solve for y, then plug that back into one of the original equations to find x. It's like solving a puzzle where you remove pieces step by step until the picture becomes clear.

When I first tackled elimination, it seemed a bit intimidating, but practice made it click. The goal is to combine the equations to remove one variable. Start by aligning the equations vertically. Look for coefficients that are opposites or can be made opposites by multiplication. For instance, with x + y = 5 and 2x - y = 4, the y terms are +y and -y, so adding the equations cancels y, giving 3x = 9. Solve for x, then substitute into the first equation to find y.

If the coefficients don't cancel directly, multiply one or both equations to create matching or opposite coefficients. For example, with 2x + 3y = 7 and 5x + 4y = 14, you might multiply the first by 5 and the second by 2 to match the x coefficients at 10. Then subtract to eliminate x and solve for y. Always verify your solution by substituting back into both original equations to ensure they hold true.