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Solve each system of equations.$$\begin{aligned}2 x-y+3 z &=14 \\x+y-2 z &=-5 \\3 x+y-z &=2\end{aligned}$$

$\{(2,-1,3)\}$

Algebra

Algebra 2

Chapter 4

Systems of Linear Equations

Section 3

Systems of Linear Equations in Three Variables

Polynomials

Systems of Equations and Inequalities

Introduction to Algebra

Graph Linear Functions

Write Linear Equations

Linear Equations and Functions

Matrices and Determinants

Johns Hopkins University

University of Texas at Austin

Oregon State University

University of Michigan - Ann Arbor

Lectures

09:38

In mathematics, an absolut…

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In mathematics, an equatio…

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Solve each system.$$\b…

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Solve the system of linear…

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02:11

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07:32

03:42

All right. So this problem are getting another system equations, Three variables. It's to ax minus y plus three z because 14 x plus y minus two z equals five reacts close y minus Z equals two. So here we do solve for one of the variables. First, plug this in salt of the other variable, plug it in salt of the third variable. So whatever to do first is I'm gonna take the second equation I'm gonna solve for X X equals, um, to the minus. Why, minus thought is now we have X in terms of Z and why? Okay, so let's plug that in to this first equation, and then we can solve for why, in terms of Z authentic sense, so we're gonna do is going to have to z minus y minus five minus. Why lost three Z equals 14. So now it for Z minus two Y minus 10 minus y plus three D because 14 this is gonna simplify. This is gonna give us seven Z minus three. Why be closed 24 like this? This is gonna give us three. Y equals 70 minus 24. Why equals 7/3 Z Okay, now we have Why, in terms of Z, we have X in terms of wine Z. So what we gonna do now is plug in this equation for why, into this equation right here. And then we have X in terms of Z and y in terms of seats. So this is gonna equal fax equals a to Z minus 7/3. Z minus eight. Minus five equals two Z minus seven. There's see. Let's be minus five. This goose in flight of three misses the same as two and one thers. So, um, this is gonna equal negative one third z with X in terms of the and y in terms of C to go leave this here. And so now we can plug this into any of the equations and actually solve for Z. Um, I'm gonna use this equation right here because we have a three Axum or multiplying by 1/3. That's gonna be an easy one to plug in three times 1/3 z firstly, plus why just seven? There's Z minus. Eat minus Z equals two is gonna give us negative Z. Well, it's not plus seven z speaks. He equals two. So it's clean this up a little bit. Nine minus eat. We know that one. We do it that the men subtract out one to make this a one. Now we have negative Z plus 7/3 Z minus dizzy. You know that Negative Z minus E is minus two Z. We have 7/3 is, um the same thing is two and 1/3. So this is just gonna give you 1/3 z equals one trains and see because three now we have That's equals three. We can play that in and solve for X and y pretty easily. It's gonna equal one plus three equals two and wise getting equal 7/3 times three is 21 divided by three, which is 77 violence eight because negative one. Now we have our order triple just a solution to the system. Equations been equal to common negative one come of three. And now you could just double check that The solution holds a few times. X four minus negative one is five plus nine. That's 14 x plus y is one minus two. Z minus six equals every five that checks out. I mean, three acts. Six plus why? Negative one minus the equals two. So these equations all hold with this solution too negative

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