Lesson 12: Equations with Infinite Solutions and Equations with No Solution #7

[One speaker]

[A white screen is seen with the words, “Solve for the variable. Determine if there is one solution, infinitely many solutions, or no solution. -4(4M - 3) = -16M + 12” written at the top.]

Instructor: Solve for the variable. Determine if there is one solution, infinitely many solutions, or no solution. So we’re going to start with our distributive property, and we can multiply negative four by both of our terms in the parenthesis. So we have negative four times four M, and then we have negative four times negative three, and that is equal to negative 16M plus twelve. [writes “-4(4M) + (-4)(-3) = -16M +12”] So negative four times four M is negative sixteen M, plus negative four times negative three is a positive twelve, and that is equal to negative sixteen M plus twelve. [writes “-16M + 12 = -16M +12”]

Now we want to try to get all of our variables on one side and our terms on the other. We want to combine like terms. So we’re going to add sixteen M to both sides. But if we notice, negative sixteen M plus sixteen M is going to cancel out and give us zero, so we’re left with twelve on one side, and on the other, we have negative sixteen M plus sixteen M, which is zero, and so we’re also left with twelve. [writes “12 = 12”] And so twelve is always equal to twelve, so that tells us that there are infinitely many solutions. It means that no matter what value we put in for M up here, it will always equal--both sides will always be equal, so there are infinitely many solutions.

Just to show that this is always the case, I’m going to solve this a little bit differently. I’m going to subtract twelve from both sides, and so that leaves us with negative sixteen M. And plus twelve minus twelve is zero, so that is just negative sixteen M on the left. And on the right, we have negative sixteen M, and we have plus twelve minus twelve, which is zero. [writes “-16M = -16M”] And now we can multiply by the multiplicative inverse--so negative one over sixteen. [adds “(-1/16)” to both sides of the equation] And we need to do that to both sides. So our sixteen’s cancel out. A negative times a negative is a positive, which leaves us with just M. Same thing on this side: our sixteen’s cancel out and a negative times a negative is positive, so we are left with M. So M is always equal to M. [writes “M = M”] So once again, that tells us there are infinitely many solutions.

[End of Video]