Question

Solve the following system. If the system's equations are dependent or if there is no solution, state this.- 4x + y + 3z = 4X + 5y + z = 45x - 4y + 3z = 0Select the correct choice below and, if necessary, fill in the answer box to complete your choice.O A. There is one solution. The solution is (0.00).(Type an exact answer in simplified form.)B. The system is dependent.O C. There is no solution.

Answer 1

Let's first copy the system of equations:

`\begin{gathered} -4x+y+3z=4 \\ x+5y+z=4 \\ 5x-4y+3z=0 \end{gathered}`

To solve the system, let's first try to reduce it to two equation with two variables.

To do this, we will start by making the substraction of the third equation by the first:

`\begin{gathered} 5x-4y+3z=0 \\ -(-4x+y+3z=4)\to4x-y-3z=-4 \\ 5x+4x-4y-y+3z-3z=0-4 \\ 9x-5y=-4 \end{gathered}`

Now, we have to do the same for another pair, let's do for the first two. However, since the second equation has a "1" as the multiplier for "z", we have to multiply this equation by 3:

`\begin{gathered} -4x+y+3z=4 \\ -3(x+5y+z=4)\to-3x-15y-3z=-12 \\ -4x-3x+y-15y+3z-3z=4-12 \\ -7x-14y=-8 \\ 7x+14y=8 \end{gathered}`

Now we have a simpler system of equations:

`\begin{gathered} 9x-5y=-4 \\ 7x+14y=8 \end{gathered}`

Let's solve for "x" in the first equation and substitute it into the second:

`\begin{gathered} 9x-5y=-4 \\ 9x=5y-4 \\ x=(5y-4)/(9) \end{gathered}`

Substitute:

`\begin{gathered} 7\mleft((5y-4)/(9)\mright)+14y=8 \\ (35y)/(9)-(28)/(9)+14y=8 \\ (35y)/(9)+14y=8+(28)/(9) \\ (35y+126y)/(9)=(72+28)/(9) \\ 161y=100 \\ y=(100)/(161) \end{gathered}`

Now that we have got a value for "y", we can substitute it into the equation that we solved for "x":

`\begin{gathered} x=(5y-4)/(9) \\ x=(5(100)/(161)-4)/(9)=((500)/(161)-(644)/(161))/(9)=(-(144)/(161))/(9)=-(144)/(161)\cdot(1)/(9)=-(16)/(161) \\ x=-(16)/(161) \end{gathered}`

With the solutions for "x" and "y", we can substitute it into any of the equations containing "z" and solving for it.

Let's do it into the second one:

`\begin{gathered} x+5y+z=4 \\ -(16)/(161)+5\cdot(100)/(161)+z=4 \\ z=4+(16)/(161)-(500)/(161) \\ z=(644+16-500)/(161) \\ z=(160)/(161) \end{gathered}`

So, the answer is that "There is one solution", and this solution is:

`\begin{gathered} x=-(16)/(161) \\ y=(100)/(161) \\ z=(160)/(161) \end{gathered}`