Well let's start with the first problem 2x + 6y = -8 is there a common factor?
make the x's the same number then multiply one eqn by -1. add the equations and solb=ve for the one not cancelled out. then plug in either equation the one known and find the other known
The common factor is 2, right Koala?
yes, good job. So if you divide everything by 2 what is left in the first problem?
x + 3y = -4
now how can we move the y to the right side?
What do I do from here? I am easily confused, sorry...
how can you move the y value to the right side?
\[2x + 6y = -8\]\[ 5x - 3y = 88\]When we work by elimination, we try to multiply one or more of the equations by a constant so that we end up with one of the variables having equal but opposite coefficients. Then we can add the equations together and one of the variables (the one with the equal but opposite coefficients) will "disappear" leaving us with a simpler problem. We may have to do this multiple times to get to the answer.
Here's a simple example of where we are trying to go: \[2x+2y = 6\]\[4x-2y=-6\] Notice that we have the coefficients of \(y\) as equal but opposite values: \(2\) and \(-2\) If we add the two equations together, look what happens: \[2x+2y = 6\]\[4x-2y=-6\]-------------- \[2x+4x+2y+(-2y)=6+(-6)\]or\[6x+0y=0\]which is equivalent to \[6x=0\]The solution to that simple equation is \(x = 0\). Now we take our value of \(x\) and we put it back in one of the original equations and find the value of \(y\): \[2x+2y=6\]\[2(0)+2y=6\]\[0+2y=6\]\[2y=6\]\[y=3\] So, the solution to that system of equations is \((0,3)\) or \(x=0,\ y=3\)
The problem we have to solve, however, is not quite as convenient. None of our variables have equal but opposite coefficients like we enjoyed in my example problem. Not to worry, though — we can make it happen! \[2x + 6y = -8\]\[5x-3y=88\] Look at the \(y\) coefficients: one is \(+6\), the other is \(-3\). Added together, they don't make \(0\), but if we add \(6\) and \(2*(-3)\), we do get \(0\) as we want. So, remembering that we can do just about anything to an equation and preserve its truth if we do that to both sides of the equation, let's multiply our second equation by \(2\): \[2x+6y=-8\]\[2*5x-2*3y=2*88\]or\[2x+6y=-8\]\[10x-6y=176\] Now we have equal but opposite coefficients for \(y\), and we can add the two equations together to get an equation that only involves \(x\), not \(x\) and \(y\). \[2x+6y=-8\]\[10x-6y=176\]------------\[2x+10x+6y-6y=-8+176\] Can you finish the simplification and solution for the value of \(x\)?