Solving Systems of Linear Equations

Solving using the Multiplication method

Ok it is time to explain how to use the multiplication method to solve equations. So I have a example problem here, Our 1st problem is 3x + 2y = 1, and out second problem is 4x + 3y = -2. Do you know what to do?

If you said multiply the 1st equation by 3 and the 2nd equation by 2 then your correct. If you are wondering why I did this, it is simple, sense we are using the multiplication method, we are going to simplify by making 2 numbers match by multiplying. In this case we want the 2 from the first equation and the 3 from the second equations (the y's) to match so we have to multiply the 1st equation by 3 and the 2nd by 2. And the bottom equations is the multiplied equations.

Now that we are here we can simplify! Sense there are two 6y's we can cancel out. Then combine the equations.

Okay I know what you are thinking, "WOAH, this is a mess"! Maybe it is, but let me explain, going from left to right. On 9x - 8x, we got 1x which can also be written as x, the y's are completely out of this part of the equation, and for the weird part, we have a positive 3- a negative 4, there is a simple way to fix this, there is two -'s it this so it turns this problem into to positives being added. So you get a positive 7. So X=7. NO, we are NOT done!

At this step you can use which ever of the equations you want to use but in my case im using 3x + 2y = 1. Because X=7, we can replace the x with 7, making the equation 3(7) + 2y = 1, multiply 3 x 7.

So after we have multiplied we have gotten 21 + 2y = 1

After that we must subtract 21 from 21, to cancel it out. But because math must be difficult, what we do to one side, we must do to the other, so 1 minus 21.

After that we get negative 20 and also 21 no longer exists in the problem, we must bring down the 2y and divide it by 2 and we must do the same to the other side, so -20 divided by 2.

After we divide -2o by 2, we get Y=-10!

Plot it like it is on a graph, in our case be have (7,-10) because X comes first.

CHECK THE WORK!!! Take one of the problems and fill in the letters with there respective numbers. In this case 3(7) + 2(-10) = 1.

Now do the math, 3 x 7 = 21 and 2 x -10 is -20, so fill in another problem in our case 21+-20 = 1. Does 21 + -20 = 1. Yes it does. So we are correct. X= 7 and Y= -10. We are done here.