Cramers Rule Calculator

Cramer's Rule calculator solve a system of three linear equations with real coefficients. It is an online algebra tool programmed to determine an ordered triple as a solution to a system of three linear equations. Using this calculator, we will able to understand the algorithm of how to solve the system of linear equations using Cramer's rule.
It is necessary to follow the next steps:

1. Enter twelve coefficients of a system of linear equations in the box. These coefficients must be real numbers.
2. Press the "CALCULATE" button to make the computation;
3. Cramers rule calculator will give an ordered triple $(x,y,z)$ as a solution of a system of three linear equations.

Input : System of three linear equations.;
Output : Three real numbers.

Cramer's Rule in Three Variables Formula: The solution of system
$$\begin{align} &a_1x+b_1y+c_1z=\color{blue}{d_1}\\ &a_2x+b_2y+c_2z=\color{blue}{d_2}\\ &a_3x+b_3y+c_3z=\color{blue}{d_3}\\ \end{align} $$ is determined by the formulas $$x=\frac{\left| \begin{array}{ccc} \color{blue}{d_1} & b_1 &c_1\\ \color{blue}{d_2} &b_2 &c_2\\ \color{blue}{d_3} &b_3 &c_3\\ \end{array} \right|}{\left| \begin{array}{ccc} a_1 & b_1 &c_1\\ a_2 &b_2 &c_2\\ a_3 &b_3 &c_3\\ \end{array} \right|},\quad y=\frac{\left| \begin{array}{ccc} a_1 & \color{blue}{d_1} &c_1\\ a_2 &\color{blue}{d_2} &c_2\\ a_3 &\color{blue}{d_3} &c_3\\ \end{array} \right|}{\left| \begin{array}{ccc} a_1 & b_1 &c_1\\ a_2 &b_2 &c_2\\ a_3 &b_3 &c_3\\ \end{array} \right|},\quad \mbox{and}\quad z=\frac{\left| \begin{array}{ccc} a_1 & b_1 &\color{blue}{d_1}\\ a_2 &b_2 &\color{blue}{d_2}\\ a_3 &b_3 &\color{blue}{d_3}\\ \end{array} \right|}{\left| \begin{array}{ccc} a_1 & b_1 &c_1\\ a_2 &b_2 &c_2\\ a_3 &b_3 &c_3\\ \end{array} \right|}$$

How to Find Unknown Variables by Cramers Rule?

The concept of the matrix determinant appeared in Germany and Japan at almost identical times. Seki wrote about it first in 1683 with his Method of Solving the Dissimulated Problems. Seki developed the pattern for determinants for $2 \times 2$, $3 \times 3$, $4 \times 4$, and $5 \times 5$ matrices and used them to solve equations. In the same year, G. Leibniz wrote about a method for solving a system of equations. This method is well known as Cramer's Rule. The determinant of a square matrix $A$ is a unique, real number which is an attribute of the matrix $A$. The determinant of the matrix $A$ is denoted by $det(A)$ or $|A|$.

Cramer's rule is a formula for the solution of a system of linear equations. It derives the solution in terms of the determinants of the matrix and of matrices obtained from it by replacing one column by the column vector of right sides of the equations. It is named by Gabriel Cramer (17041752), and the rule for an arbitrary number of unknowns is published in the paper [Cramer, G. (1750), Introduction a l'Analyse des lignes Courbes alg' ebriques" (in French). Geneva: Europeana. pp. 656--659].

To solve a system of linear equations using Cramer's rule we need to follow the next steps:

• Calculate a determinant of the square (main) matrix, $D$;
• Replace the $x^{th}$ column of the main matrix by the vector of right sides of the equations and calculate its determinant, $D_x$.
• To find the $x$ solution of the system of linear equations using Cramer's rule divide the determinant $D_x$ by the main determinant $D$;
• Repeat the previous step for each variable;

If the main determinant is zero the system of linear equations is either inconsistent or has infinitely many solutions.
Cramer's Rule in Two Variables: Let us consider the system of equations:
$$\begin{align} &a_1x+b_1y=\color{blue}{c_1}\\ &a_2x+b_2y=\color{blue}{c_2}\end{align} $$ The main determinant is $$D=\left| \begin{array}{cc} a_1 & b_1 \\ a_2 &b_2 \\ \end{array} \right|$$ and other two determinants are $$D_x=\left| \begin{array}{cc} \color{blue}{c_1} & b_1 \\ \color{blue}{c_2} &b_2 \\ \end{array} \right|\quad\mbox{and}\quad D_y=\left| \begin{array}{cc} a_1 & \color{blue}{c_1} \\ a_2 &\color{blue}{c_2} \\ \end{array} \right|$$ With help of determinants, $x$ and $y$ can be found with Cramer's rule as
$$x=\frac{D_x}{D}= \frac{\left| \begin{array}{cc} \color{blue}{c_1} & b_1 \\ \color{blue}{c_2} &b_2 \\ \end{array} \right|}{\left| \begin{array}{cc} a_1 & b_1 \\ a_2 &b_2 \\ \end{array} \right|}\quad\mbox{and}\quad y=\frac{D_y}{D}=\frac{\left| \begin{array}{cc} a_1 & \color{blue}{c_1} \\ a_2 &\color{blue}{c_2} \\ \end{array} \right|}{\left| \begin{array}{cc} a_1 & b_1 \\ a_2 &b_2 \\ \end{array} \right|}$$ If every determinant is zero, the system is consistent and equations are dependent. The system has infinitely many solutions. If $D=0$ and $D_x$ or $D_y$ is not zero, the system is inconsistent and does not have a solution.
Cramer's Rule in Three Variables: Let us consider the system of equations: $$\begin{align} &a_1x+b_1y+c_1z=\color{blue}{d_1}\\ &a_2x+b_2y+c_2z=\color{blue}{d_2}\\ &a_3x+b_3y+c_3z=\color{blue}{d_3}\\ \end{align} $$ The main determinant is $$D=\left| \begin{array}{ccc} a_1 & b_1 &c_1\\ a_2 &b_2 &c_2\\ a_3 &b_3 &c_3\\ \end{array} \right|$$ and other three determinant are $$D_x=\left| \begin{array}{ccc} \color{blue}{d_1} & b_1 &c_1\\ \color{blue}{d_2} &b_2 &c_2\\ \color{blue}{d_3} &b_3 &c_3\\ \end{array} \right|\quad D_y=\left| \begin{array}{ccc} a_1 & \color{blue}{d_1} &c_1\\ a_2 &\color{blue}{d_2} &c_2\\ a_3 &\color{blue}{d_3} &c_3\\ \end{array} \right|\quad\mbox{and}\quad D_z=\left| \begin{array}{ccc} a_1 & b_1 &\color{blue}{d_1}\\ a_2 &b_2 &\color{blue}{d_2}\\ a_3 &b_3 &\color{blue}{d_3}\\ \end{array} \right|$$ The solution of the system of three equations is $$x=\frac{D_x}{D},\quad y=\frac{D_y}{D},\quad \mbox{and}\quad z=\frac{D_z}{D}$$ For example, let us solve the system of linear equations: $$\begin{align} &3x+4y+5z=10\\ &5x+6y+7z=12\\ &4x+5y+0z=15\\ \end{align} $$ Firstly, we calculate the main determinant: $$\begin{align} D&=\left| \begin{array}{ccc} 3 & 4 &5\\ 5 &6 &7\\ 4 &5 &0\\ \end{array} \right|\&=\left|\begin{array}{ccc|cc} 3 & 4 & 5&3 & 4 \\ 5& 6 & 7&5& 6 \\ 4 &5 & 0&4&5 \\ \end{array} \right.=3\cdot6\cdot0+4\cdot7\cdot4+5\cdot5\cdot 5-5\cdot6\cdot4-3\cdot7\cdot5-4\cdot6\cdot0=12\end{align}$$ Similarly, $$ D_x=\left| \begin{array}{ccc} \color{blue}{10} & 4 &5\\ \color{blue}{12} &6 &7\\ \color{blue}{15} &5 &0\\ \end{array} \right|=-80,\quad D_y=\left| \begin{array}{ccc} 3 & \color{blue}{10} &5\\ 5 &\color{blue}{12} &7\\ 4 &\color{blue}{15} &0\\ \end{array} \right|=100,\quad D_z=\left| \begin{array}{ccc} 3 & 4 &\color{blue}{10}\\ 5 &6 &\color{blue}{12}\\ 4 &5 &\color{blue}{15}\\ \end{array} \right|=-8$$

Practice Problems for Cramer's Rule

Practice Problem 1:
Using the Cramer's rule, solve the system of equations
$$\begin{align} &2x+4y-z=-1\\ &x+3y+7z=2\\ &x+2y+z=-5\\ \end{align} $$ Practice Problem 2:
Using the Cramer's rule divide $\$20,500$ investment between the bond with a $10\%$ annual return and a bond with an $8\%$ annual return such that a combined annual return on the investments is $8.5\%$.

The Cramers rule calculator, formula, example calculation and practice problems would be very useful for grade school students of K-12 education to understand the concept of solving systems of linear equations. This concept is conceived in almost all areas of science, so it will be helpful in solving more complex problems.