How many solutions does a system of linear equations have if there are at least two?
About this video
A system of two linear equations has more than one solution, and the video reasons which statement must be true. Parallel lines give no solutions and crossing lines give exactly one, so the only way two lines share more than one point is to be the same line, which means infinitely many solutions.
You'll be able to
- list the three possible numbers of solutions for a system of two linear equations.
- explain why two lines that share more than one point must be the same line.
Key words
- system of linear equations
- Two or more line equations that must be true at the same time.
- solution
- An x and y pair that makes both equations true.
- infinitely many solutions
- Every point on the line is a solution, because both equations are the same line.
Transcript
You are solving a system of two linear equations in two variables. You have found more than one solution that satisfies the system. Which of the following statements is true? So before even reading these statements, let's just think about what's going on. So let me draw my axes here.
Let's draw my axes. So this is going to be my vertical axis. That could be one of the variables. And then this is my horizontal axis. That's one of the other variables. And maybe, for sake of convention, this could be x, and this could be y, but they're whatever our two variables are.
So it's a system of two linear equations. So if we're graphing them, each of the linear equations in two variables can be represented by a line. Now, there's only three scenarios here. One scenario is where the lines don't intersect at all. So the only way that you're going to have two lines in two dimensions that don't intersect is if they have the same slope and they have different y-intercepts.
So that's one scenario, but that's not the scenario that's being described here. They say, you have found more than one solution that satisfies the system. Here there are no solutions. So that's not the scenario that we're talking about. There's another scenario where they intersect in exactly one place.
So they intersect in exactly one place. There's one point, one xy-coordinate right over there that satisfies both of these constraints, but this also is not the scenario they're talking about. They're telling us that you have found more than one solution that satisfies the system.
So this isn't the scenario either. So the only other scenario that we can have-- we don't have parallel lines that don't intersect. We don't have lines that only intersect in one place. The only other scenario is that we're dealing with a situation where both linear equations are essentially the same constraint.
They both are essentially representing the same xy-relationship. That's the only way that I can have two lines, and this only applies to linear relationship and lines. But the only way that two lines can intersect more than one place is if they intersect everywhere.
So in this situation, we know that we must have an infinite number of solutions. So which of these choices say that? This one right here-- "there are infinitely many more solutions to the system"-- right over there.
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