How to Solve a System Using Cramer’s Rule?

Solving a system of linear equations can sometimes be tricky, but Cramer’s Rule offers a straightforward method if you have a square system (same number of equations as unknowns). Let’s break it down step-by-step.

What is Cramer’s Rule?

Cramer’s Rule is a mathematical theorem used to solve a system of linear equations with as many equations as unknowns, using determinants. It applies to systems of the form:

$AX = B$

where $A$ is the coefficient matrix, $X$ is the column matrix of variables, and $B$ is the column matrix of constants.

Step-by-Step Process

1. Write Down the System

Consider a system of two equations:

$begin{cases}
a_1x + b_1y = c_1 \
a_2x + b_2y = c_2
end{cases}$

Here, $A$ is the matrix of coefficients:

$A = begin{pmatrix}
a_1 & b_1 \
a_2 & b_2
end{pmatrix}$

$X$ is the matrix of variables:

$X = begin{pmatrix}
x \
y
end{pmatrix}$

And $B$ is the matrix of constants:

$B = begin{pmatrix}
c_1 \
c_2
end{pmatrix}$

2. Calculate the Determinant of $A$

The determinant of $A$, denoted as $|A|$, is calculated as:

$|A| = a_1b_2 – a_2b_1$

3. Formulate Matrices $A_x$ and $A_y$

Replace the first column of $A$ with $B$ to get $A_x$:

$A_x = begin{pmatrix}
c_1 & b_1 \
c_2 & b_2
end{pmatrix}$

Replace the second column of $A$ with $B$ to get $A_y$:

$A_y = begin{pmatrix}
a_1 & c_1 \
a_2 & c_2
end{pmatrix}$

4. Calculate Determinants $|A_x|$ and $|A_y|$

$|A_x| = c_1b_2 – c_2b_1$

$|A_y| = a_1c_2 – a_2c_1$

5. Solve for $x$ and $y$

Using Cramer’s Rule, solve for $x$ and $y$:

$x = frac{|A_x|}{|A|} = frac{c_1b_2 – c_2b_1}{a_1b_2 – a_2b_1}$

$y = frac{|A_y|}{|A|} = frac{a_1c_2 – a_2c_1}{a_1b_2 – a_2b_1}$

Example

Let’s solve the following system using Cramer’s Rule:

$begin{cases}
2x + 3y = 5 \
4x – y = 1
end{cases}$

1. Write Down the System
   Here, $A$, $X$, and $B$ are:

   $A = begin{pmatrix}
   2 & 3 \
   4 & -1
   end{pmatrix}$

   $X = begin{pmatrix}
   x \
   y
   end{pmatrix}$

   $B = begin{pmatrix}
   5 \
   1
   end{pmatrix}$

2. Calculate $|A|$

   $|A| = 2(-1) – 4(3) = -2 – 12 = -14$

3. Formulate $A_x$ and $A_y$

   $A_x = begin{pmatrix}
   5 & 3 \
   1 & -1
   end{pmatrix}$

   $A_y = begin{pmatrix}
   2 & 5 \
   4 & 1
   end{pmatrix}$

4. Calculate $|A_x|$ and $|A_y|$

   $|A_x| = 5(-1) – 1(3) = -5 – 3 = -8$

   $|A_y| = 2(1) – 4(5) = 2 – 20 = -18$

5. Solve for $x$ and $y$

   $x = frac{|A_x|}{|A|} = frac{-8}{-14} = frac{4}{7}$

   $y = frac{|A_y|}{|A|} = frac{-18}{-14} = frac{9}{7}$

   So, the solution to the system is $x = frac{4}{7}$ and $y = frac{9}{7}$

Conclusion

Cramer’s Rule provides a methodical approach to solve systems of linear equations using determinants. While it works best for small systems due to the complexity of calculating determinants, it is a powerful tool in linear algebra.

Citations

1. 1. Khan Academy – Cramer’s Rule
2. 2. Paul’s Online Math Notes – Cramer’s Rule
3. 3. Wolfram MathWorld – Cramer’s Rule