Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t.

x primex	equals=	3 x plus 4 y minus z3x+4yz
y primey	equals=	x plus zx+z
z primez	equals=	4 x minus 3 y plus 7 z4x3y+7z

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Part 1

Eliminate y and z from the system (by first eliminating z and then eliminating y), and solve the remaining differential equation for x. Choose the correct answer below.

A.x left parenthesis t right parenthesis equals Upper C 1 plus Upper C 2 e Superscript 4 t Baseline plus Upper C 3 e Superscript 6 t

x left parenthesis t right parenthesisx(t)equals=Upper C 1 plus Upper C 2 e Superscript 4 t Baseline plus Upper C 3 e Superscript 6 tC1+C2e4t+C3e6t

Your answer is correct.

B.x left parenthesis t right parenthesis equals Upper C 1 plus Upper C 2 t e Superscript 4 t Baseline plus Upper C 3 e Superscript negative 6 t

x left parenthesis t right parenthesisx(t)equals=Upper C 1 plus Upper C 2 t e Superscript 4 t Baseline plus Upper C 3 e Superscript negative 6 tC1+C2te4t+C3e6t

C.x left parenthesis t right parenthesis equals Upper C 1 plus Upper C 2 e Superscript negative 4 t Baseline plus Upper C 3 e Superscript negative 6 t

x left parenthesis t right parenthesisx(t)equals=Upper C 1 plus Upper C 2 e Superscript negative 4 t Baseline plus Upper C 3 e Superscript negative 6 tC1+C2e4t+C3e6t

D.x left parenthesis t right parenthesis equals Upper C 1 e Superscript 6 t Baseline plus Upper C 2 e Superscript 4 t Baseline plus Upper C 3 t e Superscript 4 t

x left parenthesis t right parenthesisx(t)equals=Upper C 1 e Superscript 6 t Baseline plus Upper C 2 e Superscript 4 t Baseline plus Upper C 3 t e Superscript 4 tC1e6t+C2e4t+C3te4t

Part 2

Now find y(t) and z(t) so that y(t), z(t), and the solution for x(t) found in the previous step are a general solution to the system of differential equations.

y left parenthesis t right parenthesisy(t)equals=enter your response here,

z left parenthesis t right parenthesisz(t)equals=enter your response here