FInd x, y, z, and w

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## 1 Answer

To use Cramer’s Rule we need to construct a 4×4 matrix based on the coefficients of the 4 variables. But we can simplify this to a 3×3 matrix by noting that z=-y because y+z=0. So z+w=-2 can be replaced by -y+w=-2. We work out the determinant associated with the new coefficient matrix for (w x y):

| 0   1   1 |

| 1   0 -1 |

| 1 -1   0 | which has the value ∆=(0(0-1)-1(0-(-1))+1(-1))=(0-1-1)=-2.

We use this value to find the variables as the common divisor.

w=

|   4   1   1 |

| -2   0 -1 | × -½

|   2 -1   0 |

Note that we omit the constant for the second equation because we already included y+z=0 when we reduced the problem to the new matrix.

Evaluating the determinant we get w=-½(4(-1)-2+2)=2.

x=

| 0   4   1 |

| 1  -2 -1 | × -½

| 1    2  0 |

x=-½(-4+4)=0.

y=

| 0   1   4 |

| 1   0 -2 | × -½

| 1 -1    2 |

y=-½(-4-4)=4

z=-4.

So the solution is (w,x,y,z)=(2,0,4,-4).

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