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

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to come up with the following notation for the x vector, such that:

`bar x = a*bar i + b*bar j`

You need to identify the vectors `bar u, bar v, bar w` , such that:

`bar u = bar i - 4 bar j`

`bar v = - 2 bar i - 5 bar j`

`bar w = -bar i + 2 bar j`

The problem provides the information that the vector `bar x` satisfies the following equation, such that:

`9 bar u - bar v + bar x = 10 bar x + bar w`

`9(bar i - 4 bar j) - (- 2 bar i - 5 bar j) + a*bar i + b*bar j = 10(a*bar i + b*bar j) + ( -bar i + 2 bar j)`

`(9 + 2 + a)*bar i + (-4 + 5 + b)*bar j = (10a - 1)*bar i + (10b + 2)*bar j`

Equating the coefficients of like parts yields:

`{(11+a = 10a - 1),(1+b = 10b+2):} => {(9a = 12),(9b = -1):}`

`{(a = 12/9),(b = -1/9):}`

Hence, evaluating the vector `bar x` , under teh given conditions, yields `bar x = (12/9)*bar i - (1/9)*bar j.`

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