Question

How do you write the augmented matrix for the system of linear equations `7x-5y+z=13, 19x=8z=10`?

Answer 1

See explanation

Explanation:

Given:

`7x-5y+z=13" "............Equation(1)`
`19x=8z=10 " "................Equation(2)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Equation 2 can be 'split' into two equations

`7x-5y+z=13" "............Equation(1)`
`19xcolor(white)(-5y+z)=10 ...............Equation(2_a)`
`color(white)(7x-5y)+8z=10 ...............Equation(2_b)`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now this is how we write the augmented matrix

`[(7,-5,1,|,13),(19,0,0,|,10),(0,0,8,|,10)]`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("However; if you meant, how do we solve this. It is as follows")`

`[(7,-5,1,|,13),(19,0,0,|,10),(0,0,8,|,10)]`
`R_3-:8 and R_2-:19`
`" " darr`

`[(7,-5,1,|,13),(1,0,0,|,10/19),(0,0,1,|,10/8)]`
`R_1-7R_2`
`" " darr`

`[(0,-5,1,|,237/19),(1,0,0,|,10/19),(0,0,1,|,10/8)]`
`R_1-:(-5)`
`" " darr`

`[(0,1,-1/5,|,-237/95),(1,0,0,|,10/19),(0,0,1,|,10/8)]`
`R_1+1/5R_3`
`" " darr`

`[(0,1,0,|,-667/380),(1,0,0,|,10/19),(0,0,1,|,10/8)]`

`[ (1,0,0,|,10/19),(0,1,0,|,-667/380) ,(0,0,1,|,5/4)]`

You will need to check this. It is very easy to go wrong!