`x + y + z = 5, x - 2y + 4z = 13, 3y + 4z = 13` Solve the system of linear equations and check any solutions algebraically.

Expert Answers

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You may use the substitution method to solve the system, hence, you need to use the first equation to write x in terms of y and z, such that:

`x + y + z = 5 => x = 5 - y - z`

You may now replace `5 - y - z` for x in equation` x - 2y + 4z = 13` , such that:

`5 - y - z - 2y + 4z = 13 => -3y + 3z = 8`

You may use the third equation, `3y + 4z = 13` , along with  `-3y + 3z = 8` equation, such that:

` -3y + 3z + 3y + 4z = 8 + 13 => 7z = 21 => z = 3`

You may replace 3 for z in equation `3y + 4z = 13:`

`3y + 12 = 13 => 3y = 1 => y = 1/3`

You may replace 3 for z and` 1/3` for y in equation `x = 5 - y - z:`

`x = 5 -1/3 - 3 => x = 2 - 1/3 => x = 5/3`

Hence, evaluating the solution to the given system, yields `x = 5/3, y = 1/3, z = 3.`

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