Linear Equations and Inequalities
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
Solving linear equations. This problem on the surface looks pretty straightforward, okay, there’s really not too much going one, but as we unfold it, something strange is going to happen.
Using our properties of math, we want to distribute our 4 and distribute our 2 through, just to sort of see what this problem is giving us. 4x minus 8, remember to distribute that 4 to both things, and then distribute the 2 in, 4x minus 2. Combining all like terms, 4x minus 8 and plus 5 becomes minus 3, 4x minus 2.
Typically in solving linear equations, we get all the xs to one side and all the other numbers to another side. What’s going to happen here is, when we do that we subtract 4x, our xs actually disappear from both sides all together. So this would actually just cancel and this would cancel, as well leaving us with -3 is equal to -2.
That doesn’t really make any sense, -3 doesn’t equal -2, no matter what we do. What this actually means is, there is no value of x that will work for this. If you look at what we have right here, we have a number minus 3 equals a number minus 2. That’s impossible, it never happens. Actually, what ends up happening is, if you ever end up with what I just call a false statement, something that doesn’t make sense, what it actually means is, there’s no solution, no value of x that we plug in here will ever make -3 equal to -2.
How your teacher has you distinguish this might be different. You could have no solution, does not exist, anything like that, but basically, you’re going to have something that doesn’t work. This is called a contradiction, because it contradicts itself. -3 cannot equal -2. No solution.