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Number of solutions to a system of equations

Sal is given three lines on the coordinate plane, and identifies one system of two lines that has a single solution, and one system that has no solution. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • leaf green style avatar for user Jonah.Barrera
    Could someone help me understand how to solve a equation then graph it if the equation is in standard form?
    (27 votes)
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  • aqualine ultimate style avatar for user Vinhee Cho
    Is there any difference between a coordinate plane and a coordinate grid?
    (14 votes)
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  • marcimus pink style avatar for user haleyshaina
    How can I tell if I graphed it right
    (5 votes)
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    • marcimus pink style avatar for user Rachel Wong
      there really are no sure signs that you have graphed "right"; just make sure that your incriments are small and equal. If you have graphed right, then the point of intersection that you get should be the coordinate (x,y) points that satisfy your system of equations. hope that helps!
      (3 votes)
  • hopper happy style avatar for user Nathan P
    How do you figure out if the problem has infinite, one, or no solutions?
    (4 votes)
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    • duskpin ultimate style avatar for user Dominic Nguyen
      (I'm going to reference parallel lines multiple times) Assuming you're talking about straight line equations (y=mx+b), if the equations have the same slope, they are either going to have no solutions, or infinite solutions. To picture this, if they have the same slope, but different y-int. they would be parallel lines, which never touch, which is why it has no solutions. Or, if they have the same slope, and same y-int, it would be the same line on top of itself, so every point on one of the lines shows up on the other line, which is why it has infinite solutions. If they have different slopes, then it will only have one solution, opposite from parallel lines, these will intersect once, and only once, which it why it will only have one solution. Sorry if this is too wordy, but hope this helps
      (13 votes)
  • piceratops ultimate style avatar for user Trần Nhật Nam
    so....correct me if i am wrong...but it seems like the system of two lines that has no solution has the same slope and has different y intercept
    (6 votes)
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  • blobby green style avatar for user jayda casiano
    what is the difference between solving systems graphically and solving algebraically
    (4 votes)
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  • aqualine ultimate style avatar for user Jack Schwitzer
    When it says to use standard form, do you reach standard form by algebraically transferring numbers until y is on the left, and x and whatever other numbers there may be are on the right?
    (3 votes)
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  • orange juice squid orange style avatar for user Jon Winder
    How would you figure out the number of solutions for systems of lines in three dimensions?
    (3 votes)
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  • duskpin ultimate style avatar for user AnotherUser
    If I were given a series of equations, say,
    2x+2y+4z = 6
    3x+6y-5z = 11
    Ax+7y-2z = 3 and then asked to give a value of A for which the equations are consistant, how would I go about that? I got this question (not the same equations but the same format) in an assignment (not one that I'm getting any NCEA credits for) and I'm a little lost. I was just wondering if anyone could give me some tips, or point me towards a video that addresses this kind of problem.
    (3 votes)
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  • leaf red style avatar for user Michael
    im probly failing but how can you identify a system?
    (3 votes)
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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.