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Problem 16

For the following exercises, solve each system by substitution.

$$

\begin{aligned} 4 x+6 y+9 z &=0 \\-5 x+2 y-6 z &=3 \\ 7 x-4 y+3 z &=-3 \end{aligned}

$$

Answer

$x=-2$

$y=-3$

$z=6$

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## Discussion

## Video Transcript

all right when I saw another system of three equations and three variables using the methods substitution. Um, I don't know, nothin like real life. U substitution to d'oh these problems here because none of these variables are actually like one acts or one wire one see? And so we're gonna have to do some division. Can make this a little bit more complicated than the other ones to do. All right, I still gotta pick which equation I'm gonna solvent for. What? Variable? Um, I think that I'll solve. Um, let me see. I guess I could solve the second equation for why. Okay, so I'm gonna do that. The reason why iss, um, that way mostly deal with positive numbers, like three plus five x plus since c and then divide everything by two. A y equals, um, three plus five x 60 over too. All right. I need to salt substitute this expression into my wise here and here. Okay, Let's do the top one. First is the top of the equation. Comes or thanks. Plus six. Why the wise three most five hex suits? Why? Prints plus six z two. And then don't forget their rest. of It's a plus nine z people zero. All right, so, um, we see two goes into the 63 times three. Here you distribute. So for X plus, three times three is nine plus three times five is 15 Eriks plus 18. The nine. Yeah, people zero, um, combined, Like terms. So I got four X here 50 next year. So that's 19 X and then a teensy and 99 C. Let's be post 27. See? Equals negative night checked nine from both sides. All right, so there's my taking it from three variables to variables that I need to do that again. Repeat that process and do it over here with this 30 equation. So it becomes nonsense, coz so, uh, seven and then x minus for times. But in our why value on six over too. Then seven plus three z equals naked three. Right? So, um, you see this to those in the or two times and distribute this negative to each of these terms. So my 78 started out so they don't be minus six. Um, minus 10 x minus 12 the and then plus three z equals three. Um, by like terms uh, at six to both sides. It's nine. Um, when you see seven x minus 10 X x and then negative 12 z plus three z is negative. Nine z All right, um, Missy out solve this for X. So next will equal nine plus nine z 90 over this signed by both sides and that could be simplified into not a lot of negative various native free. Yeah, Now that I have that I could take this expression here and substituted in for X here and on Lee have Z and my equation. We'll solve for at least five Z. So let's go ahead and write it out. We get 19 times my expression for X plus 27 z. It was negative. Nine. Um so this becomes negative. Seven minus seven plus 2070 was negative. Not now. I want Thio. Add 57 to both sides. So negative nine server all the age for years on this side and then negative 57 plus 27 disease. Easy and then divide both sides by 30. See 30 to be 1.6. Okay, that's the actual answer to that double. Check our work here a little bit before we go on to do on the wise and the and yes, so to here. All right. I don't see any glaring mistakes going on here. All right, Somebody go ahead. Confidently say that now X is equal to a lot of negative. Three miners, three times 116 Yeah. So no. Three brightness three times Once it's it's negative. 7.8, and then finally see, um I mean, why, why we're gonna use this. So why's an equal three was five fine. Seven years, six times 116 over too. So read waas a six times more, Quinn States, two negative. 13.2. All right, so here's our X. Why? And see. All right.