Question

Use Cramer’s rule to solve for y without solving for x, z and w in the system

2w + x + y + z = 3

−8w − 7x − 3y + 5z = −3

w + 4x + y + z = 6

w + 3x + 7y − z = 1

Answer 1

2w + x + y + z = 3

−8w − 7x − 3y + 5z = −3

w + 4x + y + z = 6

w + 3x + 7y − z = 1

Let D = $\begin{vmatrix} 2 & 1 & 1 & 1 \\ -8 & -7 & -3 & 5\\ 1&4&1&1\\ 1&3&7&-1\\ \end{vmatrix}$

C = $\begin{vmatrix} 3 \\ -3 \\ 6 \\ 1 \end{vmatrix}$

We have to solve for y,

By using Cramer's Rule we have

y = $\dfrac{D_y}{D}$

we can find D_y by replacing the 3^rd column of D matrix by the solution Matrix C. after doing this, we get

D_y as

D_y = $\begin{vmatrix} 2 & 1 & 3 & 1\\ -8 & -7 & -3 & 5\\ 1 & 4 & 6 & 1\\ 1 & 3 & 1& -1\\ \end{vmatrix}$

On solving, we get D = 424 and D_y = -78

hence y = -78/424

y = -0.184