Algebra 2 Section 3 1 Solving Systems of Equations by Graphing

section 31 is about solving system of equations by using the graphing method system of equations is when you have two or more equations that have the same variables here you see my equations both have and to solve that you find an ordered pair that satisfies both equations and one method to do that is by using the graphing method so I'm going to start with the first equation and graph it easiest way to do that is to isolate the put it in slope intercept form so here we have equal mx + so looking at this form can see my slope is -2 which I'm going to write as fraction -2 over 1 because that's going to be my rise over my run and my intercept is five which is an ordered pair of 0 five so I'm going to go ahead and graph that start with 0 five put dot there that's the intercept and my slope is -2 for the rise and one for the Run so I'm going to go down two to the right one down two to the right one and I'm going to graph that so there's the first graph for the second graph - = 1 again I'll isolate the this time subtracting soga = + 1 don't want negative want positive so I'm going to divide everything by negative 1 so positive = POS - 1 which means my slope is 1 over one and my slope or my intercept is the 01 so we graph 01 and then the slope is one over one or rise over run so end up here also so I'm going to graph that line we can take look at our lines and see if there's place where the lines cross and it looks like they cross about right here and that would be 21 now we can't be sure when we graph that 21 is the ual solution so we're going to go back to our original problem and plug two in and see if it works and plug one in so 2 * 2 would be 4 + 1 is 5 so it works in the first and 2 for minus 1 for yal 1 so it works in the second so we know that our ordered pair our solution of this system is the ordered pair 2 want as we continue to practice graphing we're going to use this real world scenario to show you one of the ways that systems of equations are used here we have band that's planning to record CD and their initial startup cost is500 and each CD is going to cost $4 to produce so we're going to build an equation for what the cost of their CDs are to make well each CD costs something I'm going to call that and it costs $4 times the number of CDs they produce plus an initial startup cost of $1,500 so there's my first equation it costs them four times the amount of CDs plus $1,500 and I'm going to compare that to the income that they make so another equation we could say that their income is $10 per CD now there is no starting cost for that it's just $10 per CD so this is their equation for income we're going to take both of these equations and graph them and find out when it's going to be profitable for them to make these CDs so since we're going to graph these two we need to set up labels on our graph down here so I'm going to take and I'm going to I'm going to make each one worth 50 and I'm going to jump from 100 to 200 to 300 400 to 500 and we'll call this this number is the value so this is the number of CDs because that's the same each time and on the side we're going to say dollar amount and it's either dollar amount that we cost to to make the CDs or dollar amount that we receive now this is going to end up being in the th000 so I'm going to make each one 500 so this would be th000 this will be 2,000 3,000 4,000 and 5,000 so that's how I'm going to set up my graph and now we're going to graph both of these lines this one's already in slope intercept form so we can see that the slope is four call that four over one and we can see that the intercept is500 or 0500 now it's easy enough to find 0500 that's right here so that Point's easy now doing rise of four over run of one that's difficult especially when we have units here that are so are so big so it might be better better way to go by simply plugging in an value and getting yvalue so I'm going to plug in maybe 300 for my and I'm going to just find second point that exists on this line and that's going to be easier think than graphing the intercept so this will be 1,200 plus 1500 so 2700 is the final cost if they make 300 CDs so I'm going to get 32700 well 25's there so 27 about there I'm going to graph that line on my second line my slope is 10 over one which again is two 2 small and my my my intercept is 0 0 so know this is going to start at the origin so I'm going to begin my graph right here at 0 0 but I'm going to plug another value in let's do 300 again and if can plug 300 in for can see the is 3,000 so 33,000 is my second ordered pair and I'm going to graph that so if you do the graph well enough you should notice that they cross right here and that looks at about 250 CDs and then $2,500 so it appears is if the break even point is this spot here that means any more than 250 CDs sold and they're going to make profit because the profit line actually goes up steeper than the cost line so at this point they'll start making money here you see summary of systems of equations when you graph two lines they don't have to cross the two examples that we've just done crossed and when they do we know that there's one solution and the solution exists where the two lines intersect now there's some names that we have for that we call those kind of lines consistent if they touch and we say independent if they touch in one place now those two lines could also be drawn right on top of each other and that's what you see here you see two lines that are drawn right on top of each other and if you see that there are an infinite number of solutions because the two lines touch each other everywhere and we call that consistent because they touch and independent because they touch in all the spots and the third possibility when you graph two lines is those two lines could never touch at all and we say that those lines have no solution there's no point in which they intersect and we call those kinds of lines inconsistent so here we're asked to graph the system and then describe it give the vocabulary words consistent and independent consistent and dependent or inconsistent in other words how many solutions exist when we graph these well here two equations like to write them out side by side and then solve them separate so here I'm going to isolate the by subtracting the 9x so 9x + 24 and then I'm going to divide by -6 so we'll have equals now three goes into both of those so this will be positive 3 over 2X and six goes into 24 four times so minus 4 so that's what my equation ends up being and I'll do the same thing with the other equation 6x - 4 = 16 again I'll isolate the so we'll get positive equals three Hales because two goes into both of those and then minus 4 now here's an example of problem where don't even have to graph because look at both of those equations and they're exactly the same which means I'm going to create the same line now I'll graph it just so we can get practice here -4 is my intercept and then up three to the right two of three to the right two so it'll look like that but both equations will cause that graph to be grun so we would say that there's infinite many solutions sometimes we'll say all real numbers but what we're asked in this problem is to use the vocabulary words well they touch so we know it's consistent and they touch everywhere so it's consistent dependent here's our last example you may want to pause this and try it for yourself and then come back and see the answer so here we have our solution we isolate the over here get - 34 + 3 isolate the over here and simplify - 34s xus 2 you might notice that the slopes here are the same which if you know about slopes will tell you that these are going to be parallel I've graphed them both over here to verify and you notice they don't touch at all so if we were asking for the number of solutions we would say that this this system has no Solutions now what are the vocabulary words for that well the word we're going to use if they don't touch it all is inconsistent