# If `bara`=`barx`-`bary`and `barb`=`2barx` + `5bary, express `barx`and `bary`in terms of `bara` and barb`

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to multiply the vector `bar a`  by scalar 5 such that:

`5 bar a = 5 bar x - 5 bar y`

You need to add the given vectors `5 bar a`  and `bar b`  such that:

`5bar a+bar b = 5bar x - 5bar y + 2 bar x + 5bar y`

`5bar a+bar b = 7 bar x =gt bar x = (5bar a+bar b)/7`

You need to multiply the vector `barx`  by scalar 2 such that:

`2bar a =2 bar x -2 bar y`

You need to subtract the vector `bar b`  from `2 bar a`  such that:

`2 bar a - bar b = 2 bar x - 2 bar y - 2 bar x -5 bar y`

`2 bar a - bar b = -7 bar y`

`bar b - 2 bar a = 7 bar y`

`bar y = (bar b - 2 bar a)/7`

Hence, expressing the vectors `bar x`  and `bar y`  in terms of vectors `bar a`  and `bar b`  yields: `bar x = (5bar a+bar b)/7 ; bar y = (bar b - 2 bar a)/7.`

cosinusix | College Teacher | (Level 3) Assistant Educator

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First equation is equivalent to `barx=bar a+bary`

Substitue in the second equation:

`barb=2(bara+bary)+5bary`

`barb=2bara+2bary+5bary=2bara+7bary`

Therefore `bary=(barb-2bara)/7`

and `barx=bara+bary=bara+(barb-2bara)/7=(barb+5bara)/7`

Solution `barx=(barb+5bara)/7` ; `bary=(barb-2bara)/7`