`2x + 3y + 3z = 7, 4x + 18y + 15z = 44` Solve the system of linear equations and check any solutions algebraically.

Expert Answers

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You should notice that the system is indeterminate, since the number of variables is larger than the number of equations.

`2*(2x + 3y + 3z) = 14 => 4x + 6y + 6z = 14 => 4x = 14 - 6y - 6z`

Replace `14 - 6y - 6z ` for 4x in the second equation, such that:

`4x = 44 - 18y - 15z =>  14 - 6y - 6z = 44 - 18y - 15z `

`12y + 9z = 30 => 4y + 3z = 10 => 4y = 10 - 3z => y = 5/2 - (3/4)z`

Replace back  `5/2 - (3/4)z` for y in equation `4x = 14 - 6y - 6z` , such that:

`4x = 14 - 6( 5/2 - (3/4)z ) - 6z`

`4x = 14 - 15 + (9/2)z - 6z`

`4x = -1 - (3/2)z => x = -1/4 - (3/8)z`

Hence, evaluating the solutions to the system, yields `x = -1/4 - (3/8)z, y = 5/2 - (3/4)z, z = z.`

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