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Topic D: Systems of linear equations and their solutions

Video transcript

we're told to graph this system of equations and identify the number of solutions that it has so they have the system of equations here so they want us to graph each of these equations and think a little bit about the solutions so the first equation here I'll rewrite it so I'll graph it in the same color that I write it this first equation it's already in slope-intercept form Y is equal to 3x plus 1 we see that the slope or M is equal to 3 and we see that the y intercept the y intercept here is equal to 1 so let me be clear that is also the slope I just called it M because a lot of times people say Y is equal to MX plus B so we can graph it we can look at its y-intercept the point 0 1 must be on this graph so that's the point 0 the point 0 1 right there this is the y-axis that is the x-axis and the slope is 3 that means if we move one in the positive x direction we're going to move up 3 in the positive y direction so we move one in the X direction we move up 3 if we moved 2 in the X direction we would move up 6 we would move up 6 just like that because 6 over 2 is still 3 likewise if we moved down 1 if we went negative 1 and X we would go negative 3 and Y so negative 1 negative 3 because negative 3 divided by negative 1 is still 3 if we went negative 2 in X we would go negative 6 1 2 3 4 5 6 in Y so these are all points along the line and I can connect the dots now so let me do that so let me connect the dots as best as I can this should be a line not a curve I don't have some my hand is 100% steady but I think you get let me do it a little bit better than that I think I can do a better job than that let me draw that's even worse all right last attempt I'm just throwing me off so last attempt right here the last attempt right here there you go so that's that first line right there Y is equal to 3x plus 1 so let me do the second one now so it's written in standard form right now 2y plus 4 is equal to 6x we want to get this in slope-intercept form y is equal to MX plus B so a good place to start could be to subtract this 4 from both sides that goes on the other side so let's subtract 4 from both sides of this equation the left-hand side we're left with just a 2y and then the right-hand side becomes 6x minus 4 so 2y is equal to 6x minus 4 and then to get everything in terms to solve for y we just have to divide everything by 2 so let's divide everything by 2 and we get Y is equal to 3x minus 2y is equal to 3x minus 2 so that's the second equation in slope-intercept form so same drill here the y-intercept is negative 2 so we go that's negative 1 negative 2 right there and it's slope is 3 and notice it's slope is the same as the other line so it's going to have the same inclination if we move one in the x direction we move up in up three in the Y direction 1 up 3 we go up 1 up 3 just like that if we go back 1 and X we go down 3 back 1 and X we go down 3 just like that so if we connect the dots here it'll look something like this do my best to draw a straight line so the second graph 2y plus 4 equals 6 we put it into slope intercept form and we graphed it now the whole point of this question was to identify the number of solutions that it has this system a solution to a system of equations is an x and y value that satisfy both of these equations now if there were such an x and y value that satisfy both of these equations then that x and y value would have to lie on both of these graphs because this blue line is all of the pairs of x and y's that satisfy the first equation the red line is all of the pairs of X's and Y's that satisfy the second equation so if something is going to satisfy both it's got to be on both lines when you look here are there any points that are on both lines well no these these two lines never intersect a point of intersection is a point that is common to both of these lines know if they don't intersect no intersection intersection so there is not a solution to this system equation so there is there is no solution we know that we know that because these two lines don't intersect and it will you didn't even have to graph it the the kind of giveaway was that these are two different lines they have different y-intercepts but their slopes are identical so if you have two different y-intercepts and your slopes are identical then you have two different lines that will never intersect and if they represent a system or if their the graphs of the system of equations that system has no solution