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Solve each system of equations.$$\begin{aligned}&x+y+z=6\\&x-y+z=2\\&x-y-z=-4\end{aligned}$$

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

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

University of California, Berkeley

University of Washington

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Oregon State University

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Okay, So here's their system operations. These three variables three announced. So we know we can solve this using substitution. However, this doesn't give us one equation with less than three variables. All equations have three variables. So what we have to do here, this all for one variable in terms of the other two. Plug in that substitution into the second equation salt for one of the variables. And then you have to go back and do it again. Um, that didn't make sense. Would go through step by step until makes more sense as we do it. So first, let's use this spirit equation and solve for X In terms of y and Z, it's gonna give us ax equals why, plus Z minus four. So now we have equation an equation for acts in terms of why and Z we can plug that into this equation and then solve for just why in terms of C So we're gonna have acts which who knows why, plus a C minus four minus y plus Z equals two. And if we saw bits for why we're going to get I don't see, we have Oh, sorry. If these wise they're gonna actually cancel. Then we're gonna be left with free with that four over. That'll kind of six. This is gonna be to Z equals six Z. It's just gonna be equal to three. Okay, so now what we can do is plug in the Z into this equation right here. One for acts, right? This is gonna give us why. Plus three minus fourth war lime on this one. Now where you could lose, you have act. You have Z. You have what X is in terms of. Why, So you can plug that into this first equation or really any of them? Let's use that first equities Another line minus one. That's why. What does he usually knows? Three Equal. Six. This is gonna give us to why, to people six or two equals four. So why is gonna equal to? And now we have why we could get X Another X equals y minus one equals two. Nice one. Because one now we have our orders. Triple external. Why? Policy? People's one comma. Two out of three. And now let's plug this in just to make sure that this solution actually works, we have X plus y plus equal +61 plus two is three plus +36 x one minus y. That's going negative. One plus Z plus three equals two. So that works now if X minus y my next Z one minus two minus three equals negative four. So the solution holds.

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