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Problem 26

For the following exercises, solve each system by Gaussian elimination.
\begin{aligned} 3 x+2 y-5 z &=6 \\ 5 x-4 y+3 z &=-12 \\ 4 x+5 y-2 z &=15 \end{aligned}

$(0,3,2)$

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## Video Transcript

All right. So another galaxy in the elimination, um, I'm gonna pick, uh, excess again to make the same. So I need a number that 35 and four we'll all go into evenly. Um, you see five on the train. Wow. It looks like you have to go all over 60. All right, So the top equation, it's mortified. 20 second equation is multiplied by 12 in the last equation. It's more. Bye, James. That is me 60 X plus 40. Why, minus 100 z. The was 1 20 70 equation 60 x, um, minus or eight. What? Plus very sexy equals. Negative. 144. And then finally, the last ones. Yes. Um, plus 75. What? Minus 30? Easy. Lead. Walls 225. All right, now you see, Let's, uh, we're subtract these two equations here. Um, let's attract them down. Okay, so six months, 60 40 minus negative. 48 is 88 negative. 100 minus 36. Maybe 130 states need, and that equals 1 20 minus. Native won 44 said be two minutes. I tracked these two equations that will subtract them, huh? Okay, so. 75 minus negative. Six by 6075 minus negative. 48. That'd be 75 plus. 48 is 123. Like you know. What's that? 1 23 Why? And then Negative 30. Minister in six equals to 25. Minus native one or iss positive. 369. How? Let me see. What did you do next? What to do next? Oh, well, the top equation. Um, they're all you. So, um, I might be able to achieve that. Four paint. 1 30 60 You 64. Okay, so everything in the top equations invincible by eight. Um all right, so let's let's do that to speak in our number of smaller. Okay, so this returns 11. Why? Bye bye. 1 36 Divided by a 17 made tea. See, Equals 264. Divided by a 33. His numbers are smaller, You see? Now these are all divisible by three. That's all we got. Yeah, that's old. So 123 letter, but three. 41. Why honest 22. See equals be 69 three, huh? All right, well, you system, these numbers are not easy. numbers, too. Eso I need more like this equation 41 But this equation by 11. All right, so 11 times Warren is 451 wife, um, minus 17 times 40 warm. 697 pews 97. Why? Equals three times 41 2 53 All right. And so then I'm gonna You're gonna do this one. So 41 times 11 it's for one again. And then negative. 22 times 11 12 Oh, these are these are my CDs. Grace this. Yes. Back 23. 53. No. All right, Well, once attract these, huh? Negative to 42. Well, you have to do this. Plus six months. Seven. Where equals zero. See people zero. Okay, so then I need to find out why. So let's use this equation here, get 88. Why? You see a zero, and this whole thing is gonna be zero here. That in turn 64 2 64 Divided by 88 three. My dignity. Your deal. Why equals three? Um, And then to find X, I can pick any of these equations. So the top one here, it's like three x plus two times Why? Minus five times z six. So, um 1000. This is six. So three X people zero so X equals zero. So zero three zero is intersection. What?