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# Number of solutions to a system of equations

Video transcript

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.