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I've was asked to solve this (as homework):

$$2x + y + z = 3$$ $$4x + 2z = 10$$ $$2x + 2y = -2$$

I need to solve it with matrices and I have NO IDEA how to do so.

I need your help. thanks.

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    What have you learned so far?2012-11-27
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    Hints: Can you write Aw = b? A is a 3x3 matrix (can you write out the equations using x, y, z)? What is b? How have you learned how to find a solution given this matrix form? Of course, maybe you have just learned some row-reduced echelon form and can do it that way and a half-dozen other ways too.2012-11-27
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    It's most unusual to assign homework on a topic not studied. Have you no notes on how the teacher said to do this? Have you no textbook to refer to?2012-11-27
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    @GerryMyerson I find that to be generally untrue, depending on the discipline and instructor. ;)2012-11-27

4 Answers 4

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$\left( \begin{array}{ccc} 2 & 1 & 1 \\ 4 & 0 & 2 \\ 2 & 2 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} 3 \\ 10 \\ -2 \end{array} \right)$

Form the augmented matrix and then use the following row operations $R_2= 1/2R_2-R_1$ $R_3= R_3-R_1$, which gives the matrix as $\left( \begin{array}{ccc} 2 & 1 & 1 & 3 \\ 0 & -1 & 0 & 2 \\ 0 & 1 & -1 & -5 \end{array} \right)$

Then use the row operation $R_3 = R_3+R_2$ which gives the new matrix as $\left( \begin{array}{ccc} 2 & 1 & 1 & 3 \\ 0 & -1 & 0 & 2 \\ 0 & 0 & -1 & -3 \end{array} \right)$

Now use back substitution to get the results

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Your system of equations is equivalent to: $$ \left( \begin{array}{ccc} 2 & 1 & 1 \\ 4 & 0 & 2 \\ 2 & 2 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} 3 \\ 10 \\ -2 \end{array} \right). $$ You want to add/subtract multiples of the three rows (see Gaussian elimination) until you get to $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} a \\ b \\ c \end{array} \right). $$

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    Angel - Thanks you! very helpful.2012-11-27
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You have:

$$A=\pmatrix{2&1&1\\4&0&2\\2&2&0}, b = \pmatrix{3\\10\\-2}$$

Do you know how to use the method of Gaussian elimination? If not, there are plenty of YouTube videos that demonstrate it. If you row reduce you get:

$$\pmatrix{1\\-2\\3}$$

as your solutions.

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$$2x + y + z = 3 \tag{1}$$ $$4x + 2z = 10 \tag{2}$$ $$2x + 2y = -2 \tag{3}$$

A clever way to attack this problem is to note that we can divide through by $2$ for equation (2) to get $2x + z = 5$. Note that $2x+z$ is a part of equation (1), so we can rewrite (1) as:

$$5 + y = 3 \implies y = -2$$

Plug $y = -2$ into equation (3) to get: $$2x + 2(-2) = -2 \implies 2x = 2 \implies x = 1$$

Plug our known values for $x$ and $y$ into either equation (1) or (2) to find the value for $z$. I'll use equation (2).

$$4(1) + 2z = 10 \implies 2z = 6 \implies z = 3$$

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    Perfect answer, very clear to understand . How couldn't I think about it? thanks you very very much!2012-11-27
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    No problem, glad to help. Feel free to click the up arrow to the left of an answer if you find it helpful :)2012-11-27
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    I'd love to. But unfortunately I'm not allowed to do so duo to my low points sum..2012-11-27
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    But but but --- it says, "I need to solve it with matrices." This is a very good way to solve the system, but it doesn't answer the question as asked.2012-11-27
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    I thought it is the only way. but it doesn't - his way better. @GerryMyerson2012-11-27
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    Yes, Joe's way is better --- when it works. What do you do when it doesn't work? The matrix way *always* works. Also, if your teacher asked you to solve the system using matrices, she may not accept an answer that doesn't use them.2012-11-28
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    I agree with @Gerry. I had simply made an answer using this method for other readers coming across the question. I would still check with your teacher or professor to see what he or she would accept as a valid solution to this problem (to see if you need to use matrices or not).2012-11-28
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    @GerryMyerson Gerry, I just can't get the way of matrix. do you know about a good tutorial? I googled and still nothing.2012-11-28
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    Every Linear Algebra textbook ever written does Gaussian elimination.2012-11-28