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# Solving equations by graphing: intro

CCSS.Math:

## Video transcript

we're told this is the graph of y is equal to three-halves to the X and that's it right over there use the graph to find an approximate solution to three-halves to the x is equal to five so pause this video and try to do this on your own before we work on this together all right now let's let's work on this so they already give us a hint of how to solve it they have the graph of y is equal to three after the X they graph it right over here and this gives us a hint and especially because it's they want us to find an approximate solution that maybe we can solve this equation or approximate a solution this equation through graphing and the way we can do that is we could take each side of this equation and set them up as a function we could set y equals to each side of it so if we set y equals to the left-hand side we get Y is equal to three has to the X power which is what they originally give us the graph of that and if we set y equal to the right hand side we get Y is equal to five and we can graph that and what's interesting here is if we can find the x value that gives us the same that Y value on both of these equations well that means that those graphs are going to intersect and if I'm getting the same Y value for that x value in both of these well then that means that 3 has to the X is going to be equal to 5 and so we can look at where they intersect and get an approximate sense of what x value that is and we can see it at least over here it looks like X is roughly equal to 4 so X is approximately equal to 4 and if we wanted to and we'd be done at that point if you wanted to you could try to test it out you could say hey does that actually work out three-halves to the fourth power is that equal to 5 let's see 3 to the fourth is 81 2 to the fourth is 16 it gets us it gets us pretty close to 5 16 times 5 is 80 so it's not exact but it gets us pretty close and if you had a graphing calculator that could really zoom in and zoom in and zoom in you would get a value you would see that X is slightly different than X equals four but let's do another example the key here is that we can approximate solutions to equations through graphing so here we are told this is the graph of y is equal to so we have this third degree polynomial right over here use the graph to answer the following questions how many solutions does the equation X to the third minus 2x squared minus X plus 1 equals negative 1/2 pause this video and try to think about that when we think about solutions to this we could say all right well let's imagine two functions one is y is equal to X to the third minus 2x squared minus X plus 1 which we already have graphed here and let's say that the other equation or the other function is y is equal to negative 1 and then how many times do these intersect that would tell us how many solutions we have so that is y is equal to negative 1 and so every time they intersect that means we have a solution to our original equation and they intersect 1 2 & 3 times so this has three solutions what about the second situation how many solutions does the equation all of this business equal to have well same drill we could set y equals 2x to the 3rd minus 2x squared minus X plus 1 and then we could think about another function what if Y is equal to 2 well y equals 2 would be up over there y equals 2 and we could see it only intersects y equals all of this business once so this is only going to have one solution so the key here and I'll just write it out and these are screenshots from the exercise on Khan Academy where you'd have to type in 1 or in the previous example you would type in 4 but these are examples where you can take an equation of one variable set both sides of them independently equal to Y graph them and then think about where they intersect because the X values where they intersect will be solutions to your original equation and a graph is a useful way of approximating what a solution will be especially if you have a graphing calculator or desmos or something like that