# Let vector x = <x_1, x_2> , vector y = <y_1, y_2, y_3> and vector z = <z_1, z_2>. Say that the systems x_1 = 2y_1 + 3y_2 - 5y_3 x_2 = -3y_1 - 4y_2 + 2y_3 and y_1 = 2z_1 - 3z_2 y_2...

Let vector x = <x_1, x_2> , vector y = <y_1, y_2, y_3> and vector z = <z_1, z_2>.

Say that the systems

x_1 = 2y_1 + 3y_2 - 5y_3

x_2 = -3y_1 - 4y_2 + 2y_3

and

y_1 = 2z_1 - 3z_2

y_2 = 3z_1 - z_2

y_3 = z_1 + z_2

(a) Rewrite these linear systems as matrix equations involving vector x, vector y, vector z.

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### 1 Answer

we can write given system of equations as

`[[x_1],[x_2]]=[[2,3,-5],[-3,-4,2]][[y_1],[y_2],[y_3]]` (i)

`[[y_1],[y_2],[y_3]]=[[2,-3],[3,-1],[1,1]][[z_1],[z_2]]` (ii)

substitute `y_1,y_2, and y_3` from (ii) in (i) ,we have

`[[x_1],[x_2]]=[[2,3,-5],[-3,-4,2]][[2,-3],[3,-1],[1,1]][[z_1],[z_2]]`

`[[x_1],[x_2]]=[[4+9-5,-6-3-5],[-6-12+2,9+4+2]][[z_1],[z_2]]`

`[[x_1],[x_2]]=[[8,-14],[-16,15]][[z_1],[z_2]]`

`[[x_1],[x_2]]=[[8z_1-14z_2],[-16z_1+15z_2]]` (iii)

`x_1=8z_1-14z_2`

`x_2=-16z_1+15z_2`

(i) ,(ii), (iii) are required ans.