Lesson Solving systems of non-linear equations in three unknowns using Cramer's rule

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Solving systems of non-linear equations in three unknowns using Cramer's rule


It is widely known that the Cramer's rule is an effective tool for solving linear equations when the determinant of the coefficient matrix is not equal to zero  (see the lessons
    - HOW TO solve system of linear equations in three unknowns using determinant
    - Solving systems of linear equations in three unknowns using determinant (Cramer's rule)
under the current topic in this site).
An amazing fact is that the Cramer's rule can help you to solve even non-linear equations.  It works if the original system of non-linear equations admits a reduction to a system of linear equations by introducing new variables.  For systems of two equations in two unknowns it was described in the lesson  Solving systems of non-linear equations in two unknowns using the Cramer's rule.  The current lesson provides you examples on how it works for systems of three equations in three unknowns.
I assume that you are familiar with the two lessons referred at the beginning.

Problem 1

Solve the system of three non-linear equations in three unknowns

1%2Fx + 2%2Fy - 2%2Fz = -2,
5%2Fx + 9%2Fy - 4%2Fz = -3,
3%2Fx + 4%2Fy - 5%2Fz = -3.

Solution

Let us introduce new variables  u = 1%2Fx,  v = 1%2Fy  and  w = 1%2Fz.
Then the given system of equations is reduced to the linear one:



The coefficient matrix is  %28matrix%283%2C3%2C+1%2C2%2C-2%2C+5%2C9%2C-4%2C+3%2C4%2C-5%29%29.  Its determinant is  D = 11.  (I used the solver  Finding the Determinant of a 3x3 Matrix  in this site to calculate the determinant).
The determinant is non-zero,  hence,  the Cramer's rule is applicable  (see the two lessons referred at the beginning of the lesson).

Now,  we need to calculate the determinants of three modified matrices,  according to the Cramer's rule:

Du = det%28matrix%283%2C3%2C+-2%2C+1%2C+1%2C+-3%2C+20%2C+10%2C+-3%2C+1%2C+-1%29%29 = 22,     Dv = det%28matrix%283%2C3%2C+1%2C-2%2C-2%2C+5%2C-3%2C-4%2C+3%2C3%2C-5%29%29 = -11,     Dw = det%28matrix%283%2C3%2C+1%2C2%2C-2%2C+5%2C9%2C-3%2C+3%2C4%2C-3%29%29 = 11.

(Again,  I used the solver  Finding the Determinant of a 3x3 Matrix  to calculate the determinants).

Thus the solution is  u = Du%2FD = 22%2F11 = 2,  v = Dv%2FD = %28-11%29%2F11 = -1,  and   w = Dw%2FD = 11%2F11 = 1.

Hence,  x = 1%2Fu = 1%2F2,  y = 1%2Fv = -1,  and z = 1%2Fw = 1.

You can check it by substituting the found values into he original system of equations.

Answer.  x+=+1%2F2,  y+=+-1,  z+=+1.

Problem 2

Solve the system of three non-linear equations in three unknowns

3%2F%28x%2B2%29 + 2%2F%28y-1%29 - 1%2F%28z-3%29 = 1,
2%2F%28x%2B2%29 + 3%2F%28y-1%29 - 1%2F%28z-3%29 = 1.
1%2F%28x%2B2%29 - 1%2F%28y-1%29 + 2%2F%28z-3%29 = 3.

Solution

Let us introduce new variables  u = 1%2F%28x%2B2%29,  v = 1%2F%28y-1%29  and  w = 1%2F%28z-3%29.
Then the given system of equations is reduced to the linear one:



The coefficient matrix is  %28matrix%283%2C3%2C+3%2C2%2C-1%2C+2%2C3%2C-1%2C+1%2C-1%2C2%29%29.  Its determinant is  D = 10.  (I used the solver  Finding the Determinant of a 3x3 Matrix  in this site to calculate the determinant).
The determinant is non-zero,  hence,  the Cramer's rule is applicable  (see the two lessons referred at the beginning of the lesson).

Now,  we need to calculate the determinants of three modified matrices,  according to the Cramer's rule:

Du = det%28matrix%283%2C3%2C+1%2C2%2C-1%2C+1%2C3%2C-1%2C+3%2C-1%2C2%29%29 = 5,     Dv = det%28matrix%283%2C3%2C+3%2C1%2C-1%2C+2%2C1%2C-1%2C+1%2C3%2C2%29%29 = 5,     Dw = det%28matrix%283%2C3%2C+3%2C2%2C1%2C+2%2C3%2C1%2C+1%2C-1%2C3%29%29 = 15.

(Again,  I used the solver  Finding the Determinant of a 3x3 Matrix  to calculate the determinants).

Thus the solution is  u = Du%2FD = 5%2F10 = 1%2F2,  v = Dv%2FD = 5%2F10 = 1%2F2,  and   w = Dw%2FD = 15%2F10 = 3%2F2.

Hence,  x%2B2 = 1%2Fu = 2  and  x+=+0;
             y-1 = 1%2Fv = 2  and  y+=+3;
             z-3 = 1%2Fw = 2%2F3  and  z+=+32%2F3.

You can check it by substituting the found values into he original system of equations.

Answer.  x+=+0,  y+=+3,  z+=+32%2F3.

Problem 3

Solve the system of three non-linear equations in three unknowns

3%2Fsqrt%28x-1%29 + 1%2Fsqrt%28y-2%29 - 1%2Fsqrt%28z-3%29 = 2,
3%2Fsqrt%28x-1%29 - 1%2Fsqrt%28y-2%29 + 2%2Fsqrt%28z-3%29 = 7,
-3%2Fsqrt%28x-1%29 + 2%2Fsqrt%28y-2%29 - 1%2Fsqrt%28z-3%29 = -2%29.

Solution

Let us introduce new variables  u = 1%2Fsqrt%28x-1%29,  v = 1%2Fsqrt%28y-2%29  and  w = 1%2Fsqrt%28z-3%29.
Then the given system of equations is reduced to the linear one:



The coefficient matrix is  %28matrix%283%2C3%2C+3%2C1%2C-1%2C+3%2C-1%2C2%2C+-3%2C2%2C-1%29%29.  Its determinant is  D = -15.  (I used the solver  Finding the Determinant of a 3x3 Matrix  in this site to calculate the determinant).
The determinant is non-zero,  hence,  the Cramer's rule is applicable  (see the two lessons referred at the beginning of the lesson).

Now,  we need to calculate the determinants of three modified matrices,  according to the Cramer's rule:

Du = det%28matrix%283%2C3%2C++2%2C1%2C-1%2C+7%2C-1%2C2%2C+-2%2C2%2C-1%29%29 = -15,     Dv = det%28matrix%283%2C3%2C++3%2C2%2C-1%2C+3%2C7%2C2%2C+-3%2C-2%2C-1%29%29 = -30,     Dw = det%28matrix%283%2C3%2C++3%2C1%2C2%2C+3%2C-1%2C7%2C+-3%2C2%2C-2%29%29 = -45.

(Again,  I used the solver  Finding the Determinant of a 3x3 Matrix  to calculate the determinants).

Thus the solution is  u = Du%2FD = %28-15%29%2F%28-15%29 = 1,  v = Dv%2FD = %28-30%29%2F%28-15%29 = 2,  and   w = Dw%2FD = %28-45%29%2F%28-15%29 = 3.

Hence,  sqrt%28x-1%29 = 1%2Fu = 1;
             sqrt%28y-2%29 = 1%2Fv = 1%2F2;
             sqrt%28z-3%29 = 1%2Fw = 1%2F3.

Solve the preceding three equations for  x,  y  and  z  by squaring both sides of each.  You will get

x-1 = 1  and  x = 2,
y-2 = %281%2F2%29%5E2 = 1%2F4  and  y = 21%2F4,
z-3 = %281%2F3%29%5E2 = 1%2F9  and  z = 31%2F9.

You can check it by substituting the found values into he original system of equations.

Answer.  x+=+2,  y+=+21%2F4,  z+=+31%2F9.


My lessons in this site on determinants of  3x3-matrices and the Cramer's rule for solving systems of linear equations in three unknowns are
    - Determinant of a 3x3 matrix
    - Co-factoring the determinant of a 3x3 matrix
    - HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)
    - Solving systems of linear equations in three unknowns using determinant (Cramer's rule)
    - Solving word problems by reducing to systems of linear equations in three unknowns
    - The tricks to solve some word problems with three and more unknowns using mental math
    - Sometime two equations are enough to find three unknowns by an UNIQUE way
    - Two very different approaches to one word problem
    - Solving word problems in three unknowns by the backward method
    - Solving system of linear equation in 17 unknowns
    - Solving system of linear equation in 19 unknowns
    - OVERVIEW of LESSONS on determinants of 3x3-matrices and Cramer's rule for systems in 3 unknowns
under the current topic  Matrices, determinant, Cramer rule  of the section  Algebra-II.

My other lessons in this site on solving systems of linear equations in three unknowns are
    - Solving systems of linear equations in 3 unknowns by the Substitution method,
    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Substitution method,
    - Solving systems of linear equations in 3 unknowns by the Elimination method  and
    - BRIEFLY on solving systems of linear equations in 3 unknowns by the Elimination method
under the current topic  Matrices, determinant, Cramer rule  of the section  Algebra-II.


Use this file/link  ALGEBRA-II - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-II.

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