Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$\left\{ \begin{array} { l }
2 x + 6 y = 10 \
2 x + 3 y = 7
\end{array} \right.$$
A)(10,1)
B)(2,1)
C)(2,-6)
D)(-2,1)
E)(2,-1)

Question

Use matrices to solve the system of equations (if possible).Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.​ $$\left\{ \begin{array} { l } 
2 x + 6 y = 10 \
2 x + 3 y = 7
\end{array} \right.$$ ​
A)​​(10,1)
B)​(2,1)
C)​​(2,-6)
D)​(-2,1)
E)​​(2,-1)

Quizplus · Accepted Answer

To solve the system of equations using matrices and Gaussian elimination, we first write the augmented matrix for the system:$$\begin{align*}\left[\begin{array}{cc|c}2 & 6 & 10 \2 & 3 & 7 \\end{array}\right]\end{align*}$$Next, we aim to make the element below the leading 1 in the first column 0. Since the coefficients of $x$ in both equations are the same, we can subtract the second row from the first to eliminate $x$ from the second equation:$$\begin{align*}R2 = R2 - R1 \rightarrow \left[\begin{array}{cc|c}2 & 6 & 10 \0 & -3 & -3 \\end{array}\right]\end{align*}$$Divide the second row by -3 to get a leading 1:$$\begin{align*}R2 = \frac{R2}{-3} \rightarrow \left[\begin{array}{cc|c}2 & 6 & 10 \0 & 1 & 1 \\end{array}\right]\end{align*}$$Now, to make the coefficient of $y$ in the first row 0, we can subtract 6 times the second row from the first row:$$\begin{align*}R1 = R1 - 6R2 \rightarrow \left[\begin{array}{cc|c}2 & 0 & 4 \0 & 1 & 1 \\end{array}\right]\end{align*}$$Finally, divide the first row by 2 to get a leading 1:$$\begin{align*}R1 = \frac{R1}{2} \rightarrow \left[\begin{array}{cc|c}1 & 0 & 2 \0 & 1 & 1 \\end{array}\right]\end{align*}$$This gives us the solution $x = 2$ and $y = 1$, which corresponds to choice B.

Use Matrices to Solve the System of Equations (If Possible) {2x+6y=102x+3y=7\left\{ \begin{array} { l } 2 x + 6 y = 10 \\2 x + 3 y = 7\end{array} \right.{2x+6y=102x+3y=7​

Use Matrices to Solve the System of Equations (If Possible) $\left\{ \begin{array} { l } 2 x + 6 y = 10 \\2 x + 3 y = 7\end{array} \right.$