Question

solve the following equations by cramer's rule x+y+z=2 y+z=1 x+z=3​

Answer 1

Answer:

I think the second equation should be:

y+z=2

Otherwise the question would be simply straightforward and no need to apply Cramer’s Rule to solve.

⋆ So, now let’s solve the corrected question using Cramer’s Rule.

x+y=1

y+z=2

x+z=3

The coefficient matrix :

A=⎡⎣⎢101110011⎤⎦⎥

|A|=2

The X matrix:

X=⎡⎣⎢123110011⎤⎦⎥

|X|=2

The Y matrix:

Y=⎡⎣⎢101123011⎤⎦⎥

|Y|=0

The Z matrix:

Z=⎡⎣⎢101110123⎤⎦⎥

|Z|=4

Now,

x=|X||A|

⟹x=22

⟹x=1

y=|Y||A|

⟹y=02

⟹y=0

z=|Z||A|

⟹z=42

⟹z=2

Hence, the values of x,y and z are 1,0 and 2 respectively.

P.S. - Let me illustrate Cramer’s rule for a general system of linear equation:

Consider a system of linear equation as:

a1x+b1y+c1z=d1

a2x+b2y+c2z=d2

a3x+b3y+c3z=d3

Formation of matrices:

Coefficient matrix is formed by the coefficients of x,y and z

So,

A=⎡⎣⎢a1a2a3b1b2b3c1c2c3⎤⎦⎥

X matrix is formed by substituting the first column of coefficient matrix by constants.

So,

X=⎡⎣⎢d1d2d3b1b2b3c1c2c3⎤⎦⎥

Y matrix is formed by substituting the second column of coefficient matrix by constants.

So,

Y=⎡⎣⎢a1a2a3d1d2d3c1c2c3⎤⎦⎥

Z matrix is formed by substituting the third column of coefficient matrix by constants.

So,

Z=⎡⎣⎢a1a2a3b1b2b3d1d2d3⎤⎦⎥

Now calculating x,y and z

x=|X||A|

y=|Y||A|

z=|Z||A|

where, |.| stands for determinant of a matrix.

Step-by-step explanation:

Answer 2

Step-by-step explanation:

x+y+z=2,y+z=1,z+x=3.