Classifying Solutions of Linear Equations 04:16 minutes

Video Transcript

Transcript Classifying Solutions of Linear Equations

Bert Cutler and the high school drama club are having a Murder Mystery party after school in the auditorium. Bert loves solving mysteries, so of course he is playing the detective. Everybody is acting a little suspicious. The butler looks shady, the groundskeeper is definitely hiding something, and the rich old lady is nervously clutching her pearls! So...who-dunnit? Bert goes with his gut and accuses the BUTLER! The players show their cards, but what’s this? Not only is the butler not guilty, but no one else is guilty, either. Somebody really messed up here. There’s no solution to the mystery! The drama club players reshuffle the cards and deal again. Bert accuses the groundskeeper and this time. He's right! The groundskeeper is guilty, but all the other players are guilty too! So, what 's going on? To figure out the mystery inside this mystery, we're going to have to investigate classifying solutions to linear equations. Just like the students' murder mystery game, a linear equation can have NO solution. Let’s look at an example. To calculate the solution to a linear equation, we need to follow the proper steps. First, to get rid of the parentheses, we’ll use the Distributive Property. Then, use opposite operations to isolate the 'x'. Hmm...this answer doesn't look right. 2 is NOT equal to 0. This is a false statement, which means there’s no solution to this linear equation. So, if after following the steps to solve a linear equation, you end up with an answer that looks like something like this, and 'a' and 'b' are not the same number. That means the linear equation has no solution. But there are also cases when a linear equation has an infinite number of solutions. Just like in the game when EVERYONE was guilty, in these cases ANY number can be a solution. Let’s take a look at an example. As always, follow the proper steps to solve the linear equation. First, use the Distributive Property to eliminate the parentheses and then combine the like terms. Look! Both sides of the equal sign are the same! We could simplify that even further by subtracting 2 from both sides, then 5x from both sides, and we end up with an equation that is obviously always going to be true. This means this linear equation has infinitely many solutions. So, if after following the proper steps, you get something that looks like this, or both sides of the equal sign have the same expression. It means there are infinite solutions to the linear equation. The last kind of solution is one that you've seen lots of times: a linear equation with just one solution. By now, you're a pro at this. Use the Distributive Property to write the equation without the parentheses. Combine the like terms and use opposite operations to isolate the variable. As you can see, there's just one solution to this linear equation. To review, there are three different kinds of solutions to linear equations. A linear equation can have no solution. You can recognize this when you end up with an equation like 2 equals 0, which is NOT (never) true. A linear equation can also have an infinite number of solutions. This is when you end up with an equation that is ALWAYS true, like zero equals zero. Finally, there can also be just one solution to a linear equation. These are equations like 5 equals 'x', where a variable is equal to a number. Bert's ready to solve the mystery. This time, he's sure he can identify the guilty person because there should be just one solution to this mystery. Bert takes center stage and accuses...himself! I knew there was something suspicious about that guy!