# Let vector u=<1,-4>,vector v=<-2,-5>, and vector w=<-1,2>. Find the length of vector x that satisfies 9(|u|)-(|v|)+(|x|) = 10(|x|)+(|w|)

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You need to evaluate the lengths of the vectors `bar u, bar v, bar w` and `bar x` such that:

`|bar u| = sqrt(1^2 + (-4)^2) => |bar u| = sqrt 17`

`|bar v| = sqrt((-2)^2 + (-5)^2) => |bar v| = sqrt 29`

`|bar w| = sqrt ((-1)^2+2^2) => |bar w| = sqrt 5`

`|bar x| = sqrt(a^2 + b^2)`

The problem provides the information that `9(|u|)-(|v|)+(|x|) = 10(|x|)+(|w|)` , hence, you need to substitute the lengths evaluated above, such that:

`9sqrt17 - sqrt29 + sqrt(a^2 + b^2) = 10sqrt(a^2 + b^2) + sqrt 5`

You need to isolate the terms that contain `sqrt(a^2 + b^2)` to one side, such that:

`9sqrt(a^2 + b^2) = 9sqrt17 - sqrt29 - sqrt5`

Dividing by 9 both sides yields:

`sqrt(a^2 + b^2) = sqrt17 - sqrt29/9 - sqrt5/9`

Hence, evaluating the length of vector `bar x` , under the given conditions, yields `|bar x| = sqrt(a^2 + b^2) = sqrt17 - sqrt29/9 - sqrt5/9.`