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# Systems of equations with graphing

CCSS.Math:

## Video transcript

let's say I have the equation y is equal to X plus 3 and I want to graph all of the sets all of the coordinates X comma Y that satisfy this equation right there and we've done this many times before so we draw our axis our axes that's my Y axis that's my y-axis this is my x-axis that's my x axis and this is already in MX plus B form or slope-intercept form the y-intercept here is y is equal to 3 and the slope here is 1 so this line is going to look like this we intersect at 0 comma 3 1 2 3 at 0 comma 3 and we have a slope of 1 so every one we go to the right we go up 1 so the line will look something like that it will look something like that - good enough approximation so the line will look like this and remember when I'm drawing a line this every point on this line is a solution to this equation or it represents a pair of x and y that satisfy this equation so maybe when you take let's say you take of X is equal to when you take X is equal to 5 when X is equal to 5 you go to the line and you're going to see gee when X is equal to 5 on that line y is equal to 8 is a solution and it's going to sit on the line so this represents the solution set to this equation all of the coordinates that satisfy y is equal to X plus 3 now let's say we have another equation let's say we have an equation let's say Y is equal to negative x plus 3 and we want to graph all of the X and y pairs that satisfy this equation what we could do the same thing this has a y-intercept also at 3 right there but its slope is negative 1 so it's going to look something like this it's going to look something like this every time you move to the right one you're going to move down 1 or if you move to the right a bunch you're going to move down that same bunch so that's what this equation will look like or this is this represents all every point on this line represents a X&Y pair that will satisfy this equation now what if I were to ask you what is there an x and y pair that satisfies both of these equations is there a point or a coordinate that satisfies both equations well think about it everything that satisfies this first equation is on this green line right here and everything that satisfies this purple equation is on the purple line right there so what satisfies both well if there's a point that's on both lines or essentially a point of intersection of the lines so in this situation this point is on both lines and that's actually the y intercept the point zero three is on both of these lines so that coordinate pair or that XY pair must satisfy both equations and you can try it out when X is zero here zero plus three is equal to three when X is zero here is zero plus three is equal to three it satisfies both of these equations so what we just did in a graphical way is solve a system of equations a system of equations let me write that down system of equations and all that means is we have several equations each of them constrain our X's and Y so in this case the first one is y is equal to X plus 3 and then the second one is y is equal to negative x plus 3 this constraint it to a line in the XY plane this constrained our solution set to another line in the XY plane and if we want to know the X's and Y's that satisfy both of these it's going to be the intersection of those lines so one way to solve these systems of equations is to graph both lines both equations and then look at their intersection and that will be a solution to both of these equations in the next few videos we're going to see other ways to solve it that are maybe more mathematical and less graphical but I really want you to understand the graphical nature of solving systems of equations let's do another one let's do another one let's say we have y is equal to 3x minus 6 that's one of our equations and let's say the other equation is y is equal to negative x negative x plus 6 and just like the last video let's graph both of these let's graph both of these I'll try to do it as precisely as I can let's graph both of these there you go let me draw some so you want one two three four five six seven eight nine ten one two three four five six seven eight nine ten and then one two three four five six seven eight nine ten I should have just copied and pasted did some graph paper here but I think this will do the job so let's graph this purple equation here y-intercept is negative six so we have let me do a little slash one two three four five six so that's y is equal to negative six and then the slope is 3 so every time you move one you go up three you move to the right one your run is one your rise is one two three that's three all right one two three so the equation the line will look like this and it looks like I intersect at the point 2 comma 0 which is right 3 times 2 is 6 minus 6 is 0 so our line will look something like that right there that's that line there what about this line our y-intercept is +6 one two three four five six and our slope is negative one negative one so every time we go one to the right we go down one so you go one to the right you go down one so you go one to the right you go down one and so this will intersect this will intersect at well when Y is equal to zero X is equal to six one two three four five six so right over there so this line will look like that the graph I want to get it as exact as possible and so we're going to ask ourselves the same question what is an XY pair that satisfies both of these equations well you look at it here it's going to be this point this point lies on both lines on both lines there let's see if we can figure out what that point is just eyeballing the graph here it looks like we're at 1 2 3 , 1 2 3 it looks like this is the same point right there that this is the point 3 comma 3 I'm doing it just on inspecting my hand-drawn graph so maybe it's not the exact let's check this answer let's see if X is equal to 3 y equals 3 definitely satisfies both of these equations so if we check it into the first equation you get through you get 3 is equal to 3 times 3 minus 6 this is 9 minus 6 which is indeed 3 so 3 comma 3 satisfies the top equation let's see if it satisfies the bottom equation you get 3 is equal to negative 3 plus 6 and negative 3 plus 6 is indeed 3 so even with our hand-drawn graph we were able to inspect it and see that yes the point we were able to come up with the point 3 comma 3 and that does satisfy both of these equations so we were able to solve this system of equations when we say system of equations we just mean many equations that have many unknowns or they don't have to be but they tend to have more than one unknown and use each equation as a constraint on your variables and you try to find the intersection of the equations to find a solution to all of them in the next few videos we'll see more algebraic ways of solving these than drawing their two graphs and trying to find their intersection points