# Please help me with this system of equations. x+y+z = 1 ax+by+cz = 1 a^2x+b^2y+c^2z = 1 I dug around and found out its called a Vandermonde type and I need to use the Crammer method and find the...

Please help me with this system of equations.

x+y+z = 1

ax+by+cz = 1

a^2x+b^2y+c^2z = 1

I dug around and found out its called a Vandermonde type and I need to use the Crammer method and find the determinants, though I think I'm getting the wrong answer when I plug the components into the table.

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### 1 Answer

You need to evaluate the determinant of matrix of system, such that:

`Delta = [(1,1,1),(a,b,c),(a^2,b^2,c^2)]`

This structure is called Vandermonde's structure and you may evaluate the determinant of this type, such that:

`Delta = (b - a)(c - a)(c - b)`

If `a!=b!=c,` hence `Delta != 0` and the solutions to the system may be found using Cramer's method, such that:

`x = (Delta_x)/(Delta)`

You may form the determinant `Delta_x` , substituting the column of constant terms to the right with the column of coefficients of variable you need to evaluate, such that:

`Delta_x = [(1,1,1),(1,b,c),(1,b^2,c^2)]`

Since `Delta_x` represents a Vandermonde type determinant yields:

`Delta_x = (b - 1)(c - 1)(c - b)`

`x = ((b - 1)(c - 1)(c - b))/((b - a)(c - a)(c - b))`

Reducing duplicate factors, yields:

`x = ((b - 1)(c - 1))/((b - a)(c - a))`

You may evaluate y such that:

`y = (Delta_y)/(Delta)`

`Delta_y = [(1,1,1),(a,1,c),(a^2,1,c^2)]`

`Delta_y = (1 - a)(c - a)(c - 1)`

`y = ((1 - a)(c - a)(c - 1))/((b - a)(c - a)(c - b))`

Reducing duplicate factors, yields:

`y = ((1 - a)(c - 1))/((b - a)(c - b))`

You may evaluate z such that:

`z = (Delta_z)/(Delta)`

`Delta_z = [(1,1,1),(a,b,1),(a^2,b^2,1)]`

`Delta_z = (b - a)(1 - a)(1 - b)`

`z = ((b - a)(1 - a)(1 - b))/((b - a)(c - a)(c - b))`

Reducing duplicate factors, yields:

`z = ((1 - a)(1 - b))/((c - a)(c - b))`

**Hence, evaluating the solution to the system of equations, using Vandermonde's determinant and Cramer's method, yields **`x = ((b - 1)(c - 1))/((b - a)(c - a)) , y = ((1 - a)(c - 1))/((b - a)(c - b)) , z = ((1 - a)(1 - b))/((c - a)(c - b)).`