Use Cramer's Rule to solve the system of linear equations 2x + y + z = 2(a + b + c) bx + ay - z = 3ab - 2c ax + by + cz = a^2 + 2b^2 + 2c^2thanks in advance for the help... please, include how to...

Use Cramer's Rule to solve the system of linear equations

2x + y + z = 2(a + b + c)

bx + ay - z = 3ab - 2c

ax + by + cz = a^2 + 2b^2 + 2c^2

thanks in advance for the help...

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to form the matrix of system, using the coefficients of the variables x,y,z such that:

`A = ((2,1,1),(b,a,-1),(a,b,c))`

You need to evaluate the determinant of the matrix A such that:

`Delta = [[2,1,1],[b,a,-1],[a,b,c]]`

`Delta = 2ac + b^2 - a - a^2 + 2b - bc`

If `Delta!=0` , you may solve the system using Cramer's rule such that:

`x = (Delta_x)/Delta ; y = (Delta_y)/Delta ; z = (Delta_z)/Delta`

You need to find `Delta_x , Delta_y , Delta_z`  such that:

`Delta_x = [[2(a+b+c),1,1],[3ab-2c,a,-1],[a^2 + 2b^2 + 2c^2,b,c]]`

`Delta_x = 2ac(a+b+c) + 3ab^2 - 2bc - a^2 - 2b^2 - 2c^2 - a^3 - 2ab^2 - 2ac^2 - 2b(a+b+c)(a-1) - c(3ab-2c)`

`Delta_x = 2a^2c + 2abc + 2ac^2 + ab^2- 2bc - a^2 - 2b^2 - 2c^2 - a^3 - 2ac^2 - (2ab+2b^2+2bc)(a-1) - 3abc + 2c^2`

`Delta_x = 2a^2c- a^2 - a^3 - 2a^2b + 2ab- 3abc`

`Delta_x = 2a^2c + 2abc + 2ac^2 + ab^2- 2bc - a^2 - 2b^2 - a^3 - 2ac^2 - 2a^2b-2ab^2-2abc+2ab+2b^2+2bc - 3abc`

`Delta_x = 2a^2c - ab^2- a^2 - a^3 - 2a^2b + 2ab- 3abc`

You need to substitute the right column of terms for middle column of determinant `Delta`  to find `Delta_y` .

`Delta_y = [[2,2(a+b+c),1],[b,3ab-2c,-1],[a,a^2 + 2b^2 + 2c^2,c]]`

You need to substitute the right column of terms for the last column of determinant `Delta`  to find `Delta_z` .

`Delta_z = [[2,1,2(a+b+c)],[b,a,3ab-2c],[a,b,a^2 + 2b^2 + 2c^2]]`

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