Hello again all, Johnathan here today to explain the addition method, aka the elimination method for solving linear systems. This method is a neat and useful method for solving linear systems which I will do my best to summarize for you now. The first step to the addition/elimination method is to ensure that your equations are all in standard form, ax+by=c. The next step is to make it so that either the x or the y value in each equation have the same coefficient but opposite signs. If this is not already the case, you can multiply all numbers in one (or both) of the equations by a number that will make it happen. Once you’ve done this, the variables which now have the same coefficient and opposite signs can be crossed out and are no longer a part of the equation. Once these variables are crossed out, add together the remaining parts of both equations. But why does this work? Let’s look at a quick example: Say we have a system of equations which has already been put through step 2: 3x-2y= 5 -1(3x+y= 11) ⇾ -3x-y= -11 If we cross out the x variables which now have the same coefficient and opposite signs, we can then add the equation and solve. (-2y= 5) (-y= -11) (-2y-y)= (5-11) ⇾ -3y= -6 Another way this could be visualized is if we bring back the previously removed variables: 3x-2y= 5 -3x-y= -11 (3x-3x)+(-2y-y)= (5-11) ⇾ -3y= -6 As you can see, the x variables cancel each other out once again, in total equaling 0. Ensuring that these variables cancel out is what allows us to solve for the other variable. Either way, this limits the equation to only one variable, thus allowing us to solve for it. -3y(÷3)= (-6÷-3) y= 2 Now that you know the value of y, you can take one of the original equations and substitute the y variable for its value and thus find x, and there you have it: the elusive and beautiful Correct Answer. Hopefully. If you’d like to make sure your answer is correct, you can now substitute both values into the other original equation to make sure the math checks out. If this does not break any fundamental laws of mathematics, the math checks out and you're good. If this does break math or is 0 = 0, see below. IMPORTANT THINGS:
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