Worked example: number of solutions to equations

solve for x we have 8 times the quantity 3x plus 10 is equal to 28 X minus 14 minus 4x so like every equation we've done so far we just want to isolate all of the X's on one side of this equation but before we do that we can actually simplify each of these sides the left hand side we can multiply the quantity 3x Plus 10 times 8 so we're essentially just distributing the 8 the distributive property right here so this is the same thing as 8 times 3x which is 24x plus 8 times 10 which is 80 is equal to and over here we have 28 X minus 14 minus 4x so we can combine the 28 X and the minus 4x if you have 28 X minus 4x that is 24 X and then you have the minus 14 right over here now the next thing we can do and it's already looking a little bit suspicious but just to confirm that it's as suspicious as it looks let's try to subtract 24 X from both sides of this equation and if we do that we see that we actually remove the X's from both sides of the equation because we have a 24x there and we have a 24x there you might say hey let's put all the X's on the left-hand side so let's go to get rid of this 24x so you subtract 24 x right over there but you have to do it to the left-hand side as well and then we're left with on the left-hand side these guys cancel out you're left with just 80 is equal to is equal to these guys cancel out as well is equal to negative 14 now this looks very bizarre it's making a statement that 80 is equal to negative 14 which we know is not true this does not happen 80 is never equal to negative 14 they are just inherently in equal so this equation right here actually has no solution this has no solution there is no X value that will make 80 equal to negative 14