Using Cramer’s Rule to Solve a System of Two Equations in Two Variables
We will now introduce a final method for solving systems of equations that uses determinants. Known as Cramer’s Rule, this technique dates back to the middle of the 18th century and is named for its innovator, the Swiss mathematician Gabriel Cramer (1704–1752), who introduced it in 1750 in Introduction à l’Analyse des lignes Courbes algébriques. Cramer’s Rule is a viable and efficient method for finding solutions to systems with an arbitrary number of unknowns, provided that we have the same number of equations as unknowns.
Cramer’s Rule will give us the unique solution to a system of equations, if it exists. However, if the system has no solution or an infinite number of solutions, this will be indicated by a determinant of zero. To find out if the system is inconsistent or dependent, another method, such as elimination, will have to be used.
To understand Cramer’s Rule, let’s look closely at how we solve systems of linear equations using basic row operations.
We eliminate one variable using row operations and solve for the other. Say that we wish to solve for [latex]x[/latex]. If equation (2) is multiplied by the opposite of the coefficient of [latex]y[/latex] in equation (1), equation (1) is multiplied by the coefficient of [latex]y[/latex] in equation (2), and we add the two equations, the variable [latex]y[/latex] will be eliminated.
[latex]\begin{align}b_{2}a_{1}x+b_{2}b_{1}y&=b_{2}c_{1} \\ −b_{1}a_{2}x−b_{1}b_{2}y&=−b_{1}c_{2} \\ \hline b_{2}a_{1}x−b_{1}a_{2}x&=−b_{2}c_{1}−b_{1}c_{2}\end{align}[/latex] [latex]\begin{align}&\text{Multiply }R_{1}\text{ by }b_{2} \\ &\text{Multiply }R_{2}\text{ by }−b_{2} \\ \text{ } \end{align}[/latex]
Now, solve for [latex]x[/latex].
Similarly, to solve for [latex]y[/latex], we will eliminate [latex]x[/latex].
Solving for [latex]y[/latex] gives
Notice that the denominator for both [latex]x[/latex] and [latex]y[/latex] is the determinant of the coefficient matrix.
We can use these formulas to solve for [latex]x[/latex] and [latex]y[/latex], but Cramer’s Rule also introduces new notation:
- [latex]D:[/latex] determinant of the coefficient matrix
- [latex]{D}_{x}:[/latex] determinant of the numerator in the solution of [latex]x[/latex]
[latex]x=\frac{{D}_{x}}{D}[/latex]
- [latex]{D}_{y}:[/latex] determinant of the numerator in the solution of [latex]y[/latex]
[latex]y=\frac{{D}_{y}}{D}[/latex]
The key to Cramer’s Rule is replacing the variable column of interest with the constant column and calculating the determinants. We can then express [latex]x[/latex] and [latex]y[/latex] as a quotient of two determinants.
Cramer’s Rule for 2×2 systems
Cramer’s Rule is a method that uses determinants to solve systems of equations that have the same number of equations as variables.
Consider a system of two linear equations in two variables.
[latex]\begin{array}{c}{a}_{1}x+{b}_{1}y={c}_{1}\\ {a}_{2}x+{b}_{2}y={c}_{2}\end{array}[/latex]
The solution using Cramer’s Rule is given as
[latex]x=\frac{{D}_{x}}{D}=\frac{\left\rvert\begin{array}{cc}{c}_{1}& {b}_{1}\\ {c}_{2}& {b}_{2}\end{array}\right\rvert}{\left\rvert\begin{array}{cc}{a}_{1}& {b}_{1}\\ {a}_{2}& {b}_{2}\end{array}\right\rvert},D\ne 0;\text{ }\text{ }y=\frac{{D}_{y}}{D}=\frac{\left\rvert\begin{array}{cc}{a}_{1}& {c}_{1}\\ {a}_{2}& {c}_{2}\end{array}\right\rvert}{\left\rvert\begin{array}{cc}{a}_{1}& {b}_{1}\\ {a}_{2}& {b}_{2}\end{array}\right\rvert},D\ne 0[/latex].
If we are solving for [latex]x[/latex], the [latex]x[/latex] column is replaced with the constant column. If we are solving for [latex]y[/latex], the [latex]y[/latex] column is replaced with the constant column.
[latex]\begin{align}12x+3y&=15\\ 2x - 3y&=13\end{align}[/latex]