Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l }
5 x + 4 y + z = - 16 \
5 x - 2 y - z = - 2 \
4 x + y + 3 z = 7
\end{array}$$
A)(-1, 5, -4)
B)no solution
C)(-1, -4, 5)
D)(5, -4, -1)

Question

Use Gaussian elimination to solve the linear system by finding an equivalent system in triangular form.
-$$\begin{array} { l } 
5 x + 4 y + z = - 16 \
5 x - 2 y - z = - 2 \
4 x + y + 3 z = 7
\end{array}$$
A)(-1, 5, -4)
B)no solution
C)(-1, -4, 5)
D)(5, -4, -1)

Quizplus · Accepted Answer

To solve the system using Gaussian elimination, we aim to transform the system into an upper triangular form. Starting with the given system:1. $5x + 4y + z = -16$2. $5x - 2y - z = -2$3. $4x + y + 3z = 7$First, eliminate $x$ from equations 2 and 3 using equation 1. To do this, we can subtract equation 1 from equation 2 and multiply equation 1 by $\frac{4}{5}$ and subtract it from equation 3 to eliminate $x$.After performing these operations, we get a new system (simplified for clarity):1. $5x + 4y + z = -16$2. $-6y - 2z = 14$3. $-3y + 2z = 15$Next, we eliminate $y$ from equation 3 using equation 2. We can multiply equation 2 by $\frac{1}{2}$ and add it to equation 3 to eliminate $y$.This results in:1. $5x + 4y + z = -16$2. $-6y - 2z = 14$3. $z = 5$Now, we can back-substitute $z = 5$ into the previous equations to find $y$ and $x$.Substituting $z = 5$ into equation 2:$-6y - 2(5) = 14$$-6y = 24$$y = -4$Finally, substitute $z = 5$ and $y = -4$ into equation 1:$5x + 4(-4) + 5 = -16$$5x - 16 + 5 = -16$$5x = -5$$x = -1$Thus, the solution to the system is $x = -1$, $y = -4$, $z = 5$, which corresponds to choice C.

Use Gaussian Elimination to Solve the Linear System by Finding 5x+4y+z=−165x−2y−z=−24x+y+3z=7\begin{array} { l } 5 x + 4 y + z = - 16 \\5 x - 2 y - z = - 2 \\4 x + y + 3 z = 7\end{array}5x+4y+z=−165x−2y−z=−24x+y+3z=7​

Use Gaussian Elimination to Solve the Linear System by Finding $\begin{array} { l } 5 x + 4 y + z = - 16 \\5 x - 2 y - z = - 2 \\4 x + y + 3 z = 7\end{array}$