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Video transcript

solve the following application problem using three equations with three unknowns and they tell us the second angle of a triangle is fifty degrees less than four times the first angle the third angle is 40 degrees less than the first find the measures of the three angles so let's draw ourselves a triangle here let's say that the let's call the first angle a the second angle B and then the third angle C and before we even look at these constraints one property we know of triangles is that the sum of their angles must be 180 degrees so we know that a plus B plus C must be equal to 180 degrees now with that out of the way let's look at these other constraints so they tell us the second angle the second angle of a triangle let me do that in another color they tell us the second angle of a triangle is 50 degrees less than four times the first angle so we're saying B is the second angle so they're saying the second angle of a triangle is fifty degrees less than four times the first angle so four times the first angle would be 4a we're calling a the first angle so four times the first angle is 4a but it's fifty degrees less than that so minus minus 50 now the next constraint they give us the third angle is 40 degrees less than the first so the third angle is 40 degrees less than the first so the first angle is a and it's going to be 40 degrees less than that so we have three equations with three unknowns and so we just have to solve for and let's see what's the good first variable to try to eliminate and just to try to visualize that a little bit better I'm going to bring I'm going to bring these A's on to the left-hand side of each of these equations over here so I'm going to rewrite the first equation we have a plus B plus C is equal to 180 and then this equation if we subtract 4a from both sides of this equation if we subtract 4a we have negative 4 a plus B is equal to net negative is equal to negative 50 and then this equation right over here if we subtract a from both sides we get negative a negative a pull negative a plus C negative a plus C is equal to negative 40 I just subtracted a from both sides so we now want to eliminate variables and we already have this third equation here is only in terms of a and C this is only in terms of a and B and this first one is terms of a B and C so if we could let's see this is already in terms of a and see if we could turn these first two equations if we can use the information in these first two equations to end up with an equation that's only in terms of a and C then we could use whatever we end up with along with this third equation right over here and we'll have a system of two equations with two unknowns so let's do that so if we wanted to just end up with an equation only in terms of a and C using these first two we would want to eliminate the bees so we could multiply one of these equations times negative one and one of these positive bees would turn into a negative B so let's do that let's multiply this first equation over here let's multiply it times negative one times negative one so it'll become negative a minus B minus C is equal to negative 180 and then we have this green equation right over here which is really just this equation just rearranged so we have negative 4a plus B is equal to negative 50 and now we can add these two equations actually should have done that in the actually let me do it at the other color just so you see where that's coming from do it in that green color so this is negative 4 a plus B is equal to negative 50 we can add these two up now and we get we get negative a minus 4a is negative 5a the B's cancel out we have a minus C is equal to negative 180 minus 50 is negative 230 so now using these top two equations we have an equation only in terms of a and C we have another equation only in terms of a and C it looks like if we add them together their C's will cancel out so let me just rewrite this equation over here and you have to be careful that you're using all of the equations otherwise you'll kind of do a circular argument you have to be careful that over here this first equation came from these two over here now I want to combine that with this third constraint a constraint that's not already baked into this equation right over here so we have negative a plus C is equal to negative 4t and we add these two equations negative 5a minus a is negative 6a the C's cancel out and then you have negative 230 minus 40 this is equal to negative 270 we can divide both sides by negative 6 by negative 6 and we get a is equal to negative 270 over 6 let me see how many times let me see something 270 is divisible by both 3 & 2 so it should be divisible by 6 so let me just divide it the negative signs obviously will a negative divided by a negative is going to be a positive if we take 6 into 276 goes into 27 4 times 4 times 6 is 24 we subtract we get 3 bring down the 0 6 goes into 35 times so we get a is equal to a is equal to a is equal to 45 now let's look at the other ones we can substitute back in to solve for C C is equal to a minus 40 degrees a minus 40 degrees so that is equal to let me write it right over here in yellow so C is equal to 45 minus 40 which is equal to 5 degrees so so far we have a is equal to 45 degrees C is equal to 5 degrees C is equal to 5 degrees and then you could substitute into either one of these other ones to figure out B we could use this one right over here in green V is equal to 4a minus 50 so B is going to be equal to 4 times 45 is let's see 2 times 45 is 90 so 4 times 45 is 180 so it's going to be 100 eighty 180 minus 50 by this equation right over here which is equal to 130 degrees so we get B is equal to 130 degrees and then we can let me write it over here so a is equal to 45 so if actuator order if I wanted to draw this triangle it would actually look something like this a is a 45 degree angle B is 130 degree angle and and C is five so it looks it'll look something like it looks something like this it will look something like this where this is a at 45 degrees B is 135 degrees and then C is five degrees and you can verify that it works one you can just add up the angles 45 plus five is fifty I'm sorry this isn't 135 it's 130 we solved it right over here 130 and this is five so when you add them all up 45 plus 130 plus five that does indeed equal 180 degrees 45 plus five is fifty plus 130 so this does definitely equal 180 so it meets our first constraint then on our second constraint B needs to be equal to 4a minus 50 well 4 times a 4 times a is 180 minus 50 is 130 degrees so it meets our second constraint and then our third constraint C is a minus 40 degrees well a is 45 C is 5 so if you subtract 40 from 45 you get 5 which is C so it meets all of our constraints and we are done