Number of solutions to systems of equations
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We're told to look at the coordinate grid above. I put it on the side here. Identify one system of two lines that has a single solution. Then identify one system of two lines that does not have a solution. So let's do the first part first. So a single solution. And they say identify one system, but we can see here there's actually going to be two systems that have a single solution. And when we talk about a single solution, we're talking about a single x and y value that will satisfy both equations in the system. So if we look right here at the points of intersection, this point right there, that satisfies this equation y is equal to 0.1x plus 1. And it also satisfies, well, this blue line, but the graph that that line represents, y is equal to 4x plus 10. So this dot right here, that point represents a solution to both of these. Or I guess another way to think about it, it represents an x and y value that satisfy both of these constraints. So one system that has one solution is the system that has y is equal to 0.1x plus 1, and then this blue line right here, which is y is equal to 4x plus 10. Now, they only want us to identify one system of two lines that has a single solution. We've already done that. But just so you see it, there's actually another system here. So this is one system right here, or another system would be the green line and this red line. This point of intersection right here, once again, that represents an x and y value that satisfies both the equation y is equal to 0.1x plus 1, and this point right here satisfies the equation y is equal to 4x minus 6. So if you look at this system, there's one solution, because there's one point of intersection of these two equations or these two lines, and this system also has one solution because it has one point of intersection. Now, the second part of the problem, they say identify one system of two lines that does not have a single solution or does not have a solution, so no solution. So in order for there to be no solution, that means that the two constraints don't overlap, that there's no point that is common to both equations or there's no pair of x, y values that's common to both equations. And that's the case of the two parallel lines here, this blue line and this green line. Because they never intersect, there's no coordinate on the coordinate plane that satisfies both equations. So there's no x and y that satisfy both. So the second part of the question, a system that has no solution is y is equal to 4x plus 10, and then the other one is y is equal to 4x minus 6. And notice, they have the exact same slope, and they're two different lines, they have different intercepts, so they never, ever intersect, and that's why they have no solutions.