Algebra 1 Solving Systems of Equations by Graphing

hi everyone welcome to this video today where i'm teaching you how to solve system of equations by graphing this is one of the three methods we could use for solving system there's graphing substitution and elimination and using this method for graphing is always super helpful when you have graph paper available to you you can or you can easily sketch coordinate plane on piece of paper and graph your equations and see where they intersect and that's what we're going to be learning today is that the intersection is everything that's where your solution is so first of all system means that you're dealing with two or more equations and right now in algebra one we just deal with two equations later on in algebra two you're going to be dealing with three or more even so solution of system is the ordered pair or pairs that all equations in the system have in common and the way you're going to see two equations actually have an ordered pair in common is if those two lines actually intersect and that intersection is an ordered pair that is true for both equations in the system but we're going to see we know that not all lines intersect each other right so that's going to be something that definitely comes up we've got some terminology here that we need to make sure we understand we've got consistent and independent consistent and dependent and then just inconsistent so first of all if say the word consistent consistent is going to mean that there is solution so let's say go ahead give you system of equations we solve the system and you find that there is solution well we then refer to it as consistent and after we say it's consistent we can then talk about how many solutions there are so we say system is consistent and independent if there is simply just one solution we are going to also learn that you can have be consistent and dependent consistent still means there's solution but dependent is going to mean that there's infinitely many solutions so i'm gonna ask that you kind of like think about it like what would that have to mean for those two lines if there's solution and there's actually infinitely many solutions and we're going to see that in just little bit and the last one is inconsistent so think about it if consistent means there is solution then inconsistent means that there is no solution okay so we're going to take look at the first case first and we're going to take look at two problems of each of these sets so what consistent and independent looks like what it means to have system that there is solution and there's only one solution so here's our first system so this first system is equals 2x minus one and have it color coded in gray and green so i'm just going to probably change it to red just so it's little easier to see my graphs so to graph equals 2x minus 1 know plot my y-intercept at 1 my slope at 2. okay so this is also little review for our graphing skills so if you have found that you've been little rusty at graphing then this is definitely going to be helpful for you and then we're going to go ahead and connect them to make that line okay then in green i'm going to go ahead and graph the second equation equals negative three so we should remember that means on the y-axis at negative three it is horizontal line okay and now i've graphed my system and so what we should then see is well this system has an intersection okay the lines are definitely intersecting and the point that they intersect at is actually our solution so these lines are intersecting my solution is actually the coordinates of this point and you can see this point is at negative 1 negative 3 and that is the solution to my system and what this actually means negative 1 negative 3 is this solution gives you true statement for both equations so if went ahead and plugged in negative 3 for and negative 1 for look what's going to happen 2 times negative 1 is negative negative 1 negative 2 minus 1 is negative 3 it gives me true statement here it's just equals negative 3 so that means just simply plug in my value of negative 3 and notice negative 3 is equal to negative 3. it gives me true statement this is the only ordered pair that would give me true statement for both of these equations let's take look at this next one so now have equals plus 1 so i'm going to just turn the first one into red so y-intercept of one slope of one you get the point sometimes it can be little tricky making lines that looks good so now i'm going to go ahead and i'm going to graph my green line so equals negative minus 3. so have y-intercept here at negative 3 slope of negative 1. you can already see that these two lines are definitely intersecting each other which is fantastic i'm going to change my line to green i'm going to graph this line goes right through my screen box here can clearly see that these two graphs are intersecting each other and the coordinates of that intersection are my solution so these lines are intersecting the solution is negative 2 negative 1. this is what consistent and independent looks like guys if was to substitute negative 2 and negative 1 in here so it's is negative 1 is negative 2 and notice negative 2 plus 1 is negative 1. it's true statement if go ahead here and plug in negative 1 for and this would be negative negative 2 minus 3 which really means positive 2 minus 3 which is definitely negative 1 and that gives us our true statements now the next two problems are going to be consistent and dependent which means all real solutions so if go ahead and graph this first equation equals 2x minus 1. have y-intercept at negative 1 slope of 2. this is just like the first graph that we did before and graph my line then i'm going to switch over to green now should notice also here too guys 2x minus equals 1 is in standard form can find my intercepts and graph or could rearrange that equation into slope intercept form which may also help me graph it so would need to subtract 2x on both sides and then multiply the entire equation by negative 1 so end up getting an equation in slope-intercept form and then can go ahead and actually graph this equation so my y-intercept in this equation is negative one my slope is two what do we notice this equation is identical to this equation okay and if go ahead and make line of that equation obviously the lines are going to overlap each other so every point on the red line corresponds with every point on the green line and so these lines are exactly the same which means our solution is infinitely many so any point obviously for one equation here is going to be identical if plug it into the other equation they're all going to work you have infinitely many solutions so this next problem you're going to see follows that same idea have my first graph here equals negative 2x plus 5. i'm going to put that in red okay so intercept of 5 slope of negative 2 we get the point and then when go ahead and look at my second equation which is in standard form if wanted to rearrange this in slope intercept form would subtract 2x and what i'll notice is this equation is identical to the other equation think you get the point for this don't need to graph the green line over the red line and that is again infinitely many so when you have lines that intersect there is one solution if they are actually the exact same line if two equations are the exact same line that is when they are infinitely many infinitely many solutions okay let's take look at this one now so equals 2x minus 1 and equals 2x plus 1. now hopefully could only hope that you know you learned very important lesson recently where you learned about slopes and we can see that these two equations have the same slope so hopefully you're already thinking about something that you know is going to happen with these lines okay so i'm going to switch over and i'm going to graph my line through those points i'm going to switch over to green and graph my second intercept of 1 slope of 2. hopefully we see that these lines are definitely parallel and we could tell that they were parallel before we even graphed them because remember parallel lines have the same slope so if go ahead and graph these lines and see they're clearly parallel will they ever intersect no they won't intersect and we learn the solution to any system is where lines intersect each other so if you have two lines and they never intersect that means there is no solution and this is when it is inconsistent okay we would call this system inconsistent and then the last one is going to be the same exact idea if was going to take these two equations and put them into slope intercept form it would be equals negative three plus four my second equation would then become equals negative three plus two we notice they have the exact same slope right and then if make graph through those points go ahead plot my second equation with y-intercept at 2. okay and then connect them see have parallel lines and if they are parallel then they have no solution and they are inconsistent that's it those are your three types they are either intersecting and have one solution they are overlapping and the exact same line and so they have infinitely many solutions or they are parallel and they have no solutions thanks for watching bye