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:
$begin{cases}
2x + 3y = 5 \
4x – y = 1
end{cases}$
- 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}$
- Calculate $|A|$
$|A| = 2(-1) – 4(3) = -2 – 12 = -14$
- 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}$
- Calculate $|A_x|$ and $|A_y|$
$|A_x| = 5(-1) – 1(3) = -5 – 3 = -8$
$|A_y| = 2(1) – 4(5) = 2 – 20 = -18$
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.