 If the only tool you have is a hammer, then you have to treat every problem like a nail. So it's helpful to have more than one method of solving problems. And so another method of solving equations is based on the fact that we can add the same thing to both sides of an equation and obtain an equation. So if I have the two equations 3x plus 5y equals 1 and 2x plus y equals 5, and since I have equality all around, I can add the left hand sides together, I can add the right hand sides together, and the results are still equal. Now it's not quite obvious why we do this, and in fact if we do so randomly, we obtain a mess. But if we add the right terms, we can simplify. So if I have 3x plus 5y equals 1 and minus 10x minus 5y equals negative 25, and for right now we're not going to worry about where these equations came from, now if I add the left hand sides together I get minus 7x, the y variables have disappeared. And if I add the right hand sides together I get minus 24, and now my equation is much easier to solve. Now while we could wait for the right equation to drop out of the sky, it's a little risky to do so, and so something that's useful to remember is that we can multiply all terms of an equation by the same number. So let's try to solve this system of equations. So what should we multiply our equations by? Well we might do a little analysis. The coefficients of x are 3 and 2. If we multiply the first equation by 2 we get, and if we multiply the second equation by negative 3 we get, and notice that the net effect of this is that our coefficients of x are equal but opposite. So if we add we get, and our x term is dropped out and we have a much simpler equation, since this equation only has one variable we can solve it easily, y equals, and this gives us half a solution. Now we could substitute y equals negative 2-13's into an equation and solve, just like we did the last time. But I don't know about you, but we'd rather not work with fractions. So let's solve for x in the same way we solve for y. So again we note that the coefficients of y are 2 and negative 3. So if we multiply the first equation by negative 3 and the second by negative 2 we get, we can then add the two equations together because now our coefficients of y are equal but opposite and our y terms will drop out, and since this equation only has one variable we can solve it easily, x equals 23-13's, and so as an ordered pair our solution is 23-13's negative 2-13's. Well let's take a look at another system. So we'll note that the coefficients of x and the two equations are 2 and 1, so we'll multiply the first equation by 1 and the second by negative 2 and add, then solve. The coefficients of y and the two equations are negative 5 and 3, so we'll multiply the first equation by 3 and the second by negative negative 5, otherwise known as 5, then add and solve. And finally we'll give our solution as an ordered pair giving the x value first and the y value second, so the solution is xy equal to 13-11's, 3-11's.