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Problem 16 Easy Difficulty

Solve each system of equations.
$4 a-2 b+8 c=30$
$a+2 b-7 c=-12$
$2 a-b+4 c=15$


The equation $0=0$ is always true.
The planes intersect at a line and so there are an infinite number of solutions.
Therefore, there are infinite numbers of solutions for the system of equations.


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Video Transcript

Okay, so we're given this set of equations, which is called a system of equations, and our task here or our goal is to solve. So in order to do that, we're going to first look at getting the coefficients. So step number one is get the coefficients and as a reminder of those air, just the numbers in front of your variables. And the variables in this case are the a, the B in the sea, that unknown value. Get those coefficients to be the same number with opposite signs. And the purpose for this is so that we can eliminate a variable. Because think about it. If you have only two variables, it's probably gonna be an agent easier equation to solve. So if I can eliminate a variable, then life gets easy. So I'm talking about getting these coefficients the numbers in front of the variables to be the same number with opposite signs, toe like a positive four in a negative for positive, three of the negative three so that we can cancel them to eliminate a variable. But before we even go there, I'm gonna ask that you throw in some invisible ones here because remember, if there's a missing coefficient on a variable, you are allowed to assume that it's a one. Then, from here, we're gonna use some colors because colors make life fun. We're going to look at the coefficients and their signs in front of each variable and then ask ourselves which one looks the easiest to get the same number with opposite signs. So, looking at your age, you can clearly see that we have, ah, 41 and a two, all of them because can become the number four or negative for by multiplying by a value, all of the bees could be comma two by multiplying by a value or a negative. To. Of course, the seas might be a little trickier to think about, but they could all become the same number. So now you're asking yourself which one is easiest because, after all, we love to take the easy route. So here I'm going to eliminate bees. Truthfully, you could do any of them, and it would not make a difference in your solution. But bees look the easiest because I'm already seeing on these 1st 2 equations. Actually, let's take a second and label these as equation A capital A couple be capital C not to be confused with the variables of lower case. Okay, so here we've got equation A that could be added with equation. Be immediately to cancel those lower case bees. So we have four a minus to B plus eight c. All we're doing is writing those two equations on top of each other, stacking them on top one A plus two B minus seven c equals negative 12. From here, I'm going to add straight down, So put your plus sign, draw the line. We are going to just add straight down. So we've got four a plus one a to give us five a negative to be plus two b will cancel positive eight C plus negative seven C or in other words, eight. Take away. Seven is positive one c and then 30 plus negative. 12 is positive. 18 and we're going to call this equation d make a little bubbly around it because it makes it fun, but also something that you were going to come back to. We have to do the same thing because we have yet to use equation. See? So we're going to use equation, See, But if I focus on the variable and it's coefficient, it's a negative one. I need to turn it into a two or another number that all of the bees could become. The easiest option is to just multiply it by a positive to, in order to turn it to a negative, to to cancel with a different equation. But if I do it to one of the variables one part of the equation, I have to do it to all parts. So we're going to take two times equations, see, and that would turn it into a negative two right here. So we're going to cancel it with a positive two on equation. Be so two times. Equation C means that I would have a four a minus to B plus eight B or eight. See Excuse me equals 30. Every park out multiplied by two equation be stays exactly the same. One a plus to be minus seven C equals negative 12. Same process. Add straight down and we've got five a again from four plus one with the A's, the bees are going to cancel and C eight plus negative seven is again. A positive one. C 30 plus negative 12 is 18. So this is now Equation E and again another little bubbly around this because our next step is to eliminate a variable again. So we're just going to keep repeating this process. Okay, so I'm looking at Equation D and E this time, when I look at the coefficient on A I have a five and a five, both positives. But it's very easy to turn one of them into a negative if I just multiply one of the equations by a negative one. So if we take negative one times equation D and add it with equation e, I'm gonna just write down what I get And to do that, it might be easier for your brain to process by just rewriting times negative one times negative one so that you remember you're multiplying every single piece by negative one. So that leaves me with negative five a minus one C equals negative 18 and then take equation eat, as is five a plus one C equals positive 18. You're gonna draw your line, add straight down, and we expect the A's to cancel. That was what our whole purpose was, But then we're a little shocked when we see negative one c plus one see canceling again. So this will leave us with zero on the left side. When everything cancels on the left, it leaves us with zero. Then we have negative 18 plus 18 on the right side, which also leaves us with zero. So this is a very special case when we're solving a system of equations. When you come to the conclusion that zero equals zero, that's clearly a true statement. And so when zero equals zero, what this indicates is that there are infinitely money solutions to the system of equations. So final answer here is a very unique one. There are infinitely money solutions to the system of equations. Normally, you come out of this within a equals of B equals in a C equals. But because we resulted in zero equals zero, that means anything is possible here. There are so many options that could solve that original system of equations that they're infinitely many solutions to this because we have these planes for these equations intersecting at a line which is infinite meaning again, infinitely many solutions to the system

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