Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$$\begin{array} { l }
4 x - 5 y - 10 = 0 \
8 x + y - 25 = 0
\end{array}$$
A) $\varnothing$
B) $\left\{ \left( \frac { 135 } { 44 } , \frac { 5 } { 11 } \right) \right\}$
C) $\left\{ \left( \frac { 85 } { 44 } , - \frac { 5 } { 11 } \right) \right\}$
D) $\left\{ \left( - \frac { 6 } { 5 } y , y \right) \right\}$

Question

Use the Gauss-Jordan method to solve the system of equations. If the system has infinitely many solutions, give the
solution with y arbitrary.
-$$\begin{array} { l } 
4 x - 5 y - 10 = 0 \
8 x + y - 25 = 0
\end{array}$$
A) $\varnothing$
B) $\left\{ \left( \frac { 135 } { 44 } , \frac { 5 } { 11 } \right) \right\}$
C) $\left\{ \left( \frac { 85 } { 44 } , - \frac { 5 } { 11 } \right) \right\}$
D) $\left\{ \left( - \frac { 6 } { 5 } y , y \right) \right\}$

Quizplus · Accepted Answer

To solve the system using the Gauss-Jordan method, we first write the augmented matrix of the system and then apply row operations to get it in reduced row echelon form (RREF). The system of equations is:$$4x - 5y = 10$$$$8x + y = 25$$The augmented matrix is:$$\left[\begin{array}{cc|c}4 & -5 & 10 \8 & 1 & 25 \\end{array}\right]$$We aim to make the matrix look like:$$\left[\begin{array}{cc|c}1 & 0 & a \0 & 1 & b \\end{array}\right]$$Where $a$ and $b$ are the solutions for $x$ and $y$ respectively.Step 1: Make the leading coefficient of the first row 1.$$\left[\begin{array}{cc|c}1 & -\frac{5}{4} & \frac{5}{2} \8 & 1 & 25 \\end{array}\right]$$Step 2: Make the leading coefficient of the second row 0 in the first column.$$\left[\begin{array}{cc|c}1 & -\frac{5}{4} & \frac{5}{2} \0 & 11 & 5 \\end{array}\right]$$By multiplying the first row by 8 and subtracting it from the second row.Step 3: Make the leading coefficient of the second row 1.$$\left[\begin{array}{cc|c}1 & -\frac{5}{4} & \frac{5}{2} \0 & 1 & \frac{5}{11} \\end{array}\right]$$Step 4: Make the second coefficient of the first row 0.$$\left[\begin{array}{cc|c}1 & 0 & \frac{135}{44} \0 & 1 & \frac{5}{11} \\end{array}\right]$$By adding $\frac{5}{4}$ times the second row to the first row.Thus, the solution is $x = \frac{135}{44}$, $y = \frac{5}{11}$, which corresponds to choice B.

Use the Gauss-Jordan Method to Solve the System of Equations 4x−5y−10=08x+y−25=0\begin{array} { l } 4 x - 5 y - 10 = 0 \\8 x + y - 25 = 0\end{array}4x−5y−10=08x+y−25=0​

Use the Gauss-Jordan Method to Solve the System of Equations $\begin{array} { l } 4 x - 5 y - 10 = 0 \\8 x + y - 25 = 0\end{array}$