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Problem: Decomposition of vector (1,2),(3,1),(8,1) [ Calculator, Method and examples ]

Solution:
Your problem `->` Decomposition of vector (1,2),(3,1),(8,1)


Here `vec A=(1,2),vec B=(3,1),vec C=(8,1)`

Form equation from vectors
`vec C = x vec A + y vec B`

So system of linear equations are
`x+3y=8`

`2x+y=1`

Solution of equations using Elimination method

Total Equations are `2`

`x+3y=8 -> (1)`

`2x+y=1 -> (2)`



Select the equations `(1)` and `(2)`, and eliminate the variable `x`.

`x+3y=8`` xx 2->````2x``+``6y``=``16```
`2x+y=1`` xx 1->````2x``+``y``=``1```

```5y``=``15`` -> (3)`




Now use back substitution method
From (3)
`5y=15`

`=>y=(15)/(5)=3`

From (1)
`x+3y=8`

`=>x+3(3)=8`

`=>x+9=8`

`=>x=8-9=-1`

Solution using Elimination method.
`x = -1,y = 3`


`x=-1,y=3`

So, `vec C = -vec (A) +3vec (B)`






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