[Music] okay so in this video we're going to be solving simultaneous equations by a method of substitution so what this means is we want to rearrange one of the equations and substitute it into the other one and this can be quicker than elimination the method we looked at in the last video if for example we already have a variable ice isolated on its own and in this case we do have an X with only a 1 coefficient so our first step is going to be rearranging this equation to find x seck if we do this we get x equals 11 minus 3 y so let me just label that equation number 3 so now we've got X rearranged on its own in terms of Y and the next step is just the substitute it straight into the second equation so let me just write sub substitute 3 into 2 so if we do this we're going to have 4 times X and X is now 11 minus 3y and we still have the minus 7 y and this equals 6 from the equation 2 so now we've got an equation only involving wives we've eliminated the variable X and we can solve this very simple it so let's just expand that these brackets yeah it's 44 minus 12 y minus 7y and this equals 6 so we can collect the terms and simplify this is going to be 19 y minus 19 more if we move this on the other side and then 44 minus 6 is 38 so that's what we got in the left hand side and the right hand side we get 19 Y and then to find y we just divide by 19 and 38 is 2 times 19 so this gives us Y is equal to 2 ok so let's go back up here for some more space and now the second task for us to do is to find out what the corresponding value of x is and we're going to do this by substituting y equals 2 into any of these equations it doesn't matter which one but we've already got X rearranged on its own so this is the easiest one so we're going to substitute y equals 2 into equation 3 I'll just write up here substitute into number three so if we do this we'll have x equals 11 minus 3 times y and y is just to say we have x equals 11 - so wait we're there 8 6 3 times 2 3 times 2 is 6 so 11 minus 6 and we get y equals 5 so together x equals 5 and y equals 2 forms the complete solution of this pair of simultaneous equations okay so let's do one more example we're going to do 3x minus y equals to 7 that's the first equation and for the second one we have 10 x plus 3 y equals -2 so these are two linear equations and we're going to solve them by substitution so let me just write this as equation 1 and this as equation 2 so just like before we want to isolate one of these variables so X or Y as any function of the other one so here we can see an equation one that we only have a coefficient of 1 in front of Y so this is going to be that simplest to rearrange for so we can move over the wire to that side and if we do that we'll get y equals 3x minus 7 and this is just by rearranging equation one so it's still equivalent to it let's label this as equation number three so now we're going to substitute y equals 3x minus 7 into equation 2 and this is just going to get rid of the variable Y so if we do this we get X 10 X plus 3y and Y is 3x minus 7 and this equals y minus 2 so we can expand out the brackets 10 X plus 9 X minus 3 times minus 7 is minus 21 and this equals minus 2 so if there's any space left over I'm just going to click terms and we'll have 19 x equals and then remove the 21 onto that side and we also get 19 so obviously just divided by 19 we get x equals to us and now let's get back up here and the second stage is just to substitute this value of x back into any of these equations and let's do it into equation number 3 this is the simplest form that we express Y and so we get y equals 3x X is just 1 so 3 times 1 minus 7 and 3 times 1 is just free so y equals 3 minus 7 and that is also known as minus 4 so x equals 1 and y equals 2 minus 4 is the solution of this pair of linear equations and if you wanted to you could go back into one of these equations and just check that these numbers do add up and they equal but this this is a solution to this set of linear equations