# Use Cramer’s Rule to Solve a $2\times 2$ System of Linear Equations

## Problem 257

Use Cramer’s rule to solve the system of linear equations
\begin{align*}
3x_1-2x_2&=5\\
7x_1+4x_2&=-1.
\end{align*}

## Solution.

Let
$A=[A_1, A_2]=\begin{bmatrix} 3 & -2\\ 7& 4 \end{bmatrix},$ be the coefficient matrix of the system, where $A_1, A_2$ are column vectors of $A$.
Let $\mathbf{b}=\begin{bmatrix} 5 \\ -1 \end{bmatrix}$ be the constant term vector. Then the system can be written as
$A\mathbf{x}=\mathbf{b},$ where $\mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$.

We form
$B_1=[\mathbf{b}, A_2]=\begin{bmatrix} 5 & -2\\ -1& 4 \end{bmatrix}$ and
$B_2=[A_1, \mathbf{b}]=\begin{bmatrix} 3 & 5\\ 7& -1 \end{bmatrix}.$

Then Cramer’s rule gives the formula for solutions
$x_1=\frac{\det(B_1)}{\det(A)} \text{ and } x_2=\frac{\det(B_2)}{\det(A)}. \tag{*}$ Thus, it remains to compute the determinants.
We have
\begin{align*}
\det(A)=\begin{vmatrix}
3 & -2\\
7& 4
\end{vmatrix}=3\cdot 4 -(-2)\cdot 7 =26.
\end{align*}
Similarly, a calculation shows that
$\det(B_1)=18 \text{ and } \det(B_2)=-38.$

Therefore by Cramer’s rule (*), we obtain
$x_1=\frac{18}{26}=\frac{9}{13} \text{ and } x_2=\frac{-38}{26}=-\frac{19}{13}.$

Let $P_4$ be the vector space consisting of all polynomials of degree $4$ or less with real number coefficients. Let...