Q. 61 - Precalculus Enhanced with Graphing Utilities

Question

Complete the proof of Cramer’s Rule for two equations containing two variables.

Step-by-Step Solution

Answer

The Cramer Rule is Proved.

Step 1. Given Information

We need to prove Cramer Rule with two equations containing two variables. 

We will solve this task by assuming 

(a) $a=0,b\ne 0,c\ne 0,d\ne 0$

(b) $a\ne 0,b=0,c\ne 0,d\ne 0$

(c) $a\ne 0,b\ne 0,c=0,d\ne 0$

(d) $a\ne 0,b\ne 0,c\ne 0,d=0$

We change into the system of equations $$\begin{gathered} ax+by=s \\ cx+dy=t \end{gathered}$$

Step 2. Use of a = 0 , b ≠ 0 , c ≠ 0 , d ≠ 0

$$\begin{gathered} by=s or y=\frac{s}{b}-----\left(1\right) \\ cx+dy=t----\left(2\right) \end{gathered}$$

From (1) and (2)

$\begin{array}{rcl}cx+d\frac{s}{b}& =& t\\ cx& =& t-\frac{ds}{b}=\frac{bt-ds}{b}\\ x& =& \frac{bt-ds}{bc}\end{array}$

Step 3. Finding x and y

Using Cramer's formula 

$\begin{array}{rcl}x& =& \frac{\left|\begin{array}{cc}s& b\\ t& d\end{array}\right|}{\left|\begin{array}{cc}0& b\\ c& d\end{array}\right|}\\ & =& \frac{sd-tb}{-bc}\\ & =& \frac{tb-sd}{bc}\end{array}$

And,

$\begin{array}{rcl}y& =& \frac{\left|\begin{array}{cc}0& s\\ c& t\end{array}\right|}{\left|\begin{array}{cc}0& b\\ c& d\end{array}\right|}\\ & =& \frac{-sc}{-bc}\\ & =& \frac{s}{b}\end{array}$'

Step 4. Use the criteria a ≠ 0 , b = 0 , c ≠ 0 , d ≠ 0

$\begin{array}{rcl}ax& =& s or x=\frac{s}{a}\\ cx+dy& =& t\end{array}$

And,

$\begin{array}{rcl}c\frac{s}{a}+dy& =& t\\ dy& =& t-\frac{cs}{a}=\frac{ta-cs}{a}\\ y& =& \frac{at-cs}{ad}\end{array}$

Step 5. Finding x and y

$$\begin{gathered} x=\frac{\left|\begin{array}{cc}s& 0\\ t& d\end{array}\right|}{\left|\begin{array}{cc}a& 0\\ c& d\end{array}\right|}=\frac{sd}{ad}=\frac{s}{a} \\ y=\frac{\left|\begin{array}{cc}a& s\\ c& t\end{array}\right|}{\left|\begin{array}{cc}a& 0\\ c& d\end{array}\right|}=\frac{at-cs}{ad} \end{gathered}$$

Step 6. Use of a ≠ 0 , b ≠ 0 , c = 0 , d ≠ 0

$$\begin{gathered} ax+by=s \\ dy=t or y=\frac{t}{d} \end{gathered}$$

$$\begin{gathered} ax+b\frac{t}{d}=s \\ ax=s-\frac{bt}{d}=\frac{sd-bt}{d} \\ x=\frac{sd-bt}{ad} \end{gathered}$$

Step 7. Finding x and y

$$\begin{gathered} x=\frac{\left|\begin{array}{cc}s& b\\ t& d\end{array}\right|}{\left|\begin{array}{cc}a& b\\ 0& d\end{array}\right|}=\frac{sd-tb}{ad} \\ y=\frac{\left|\begin{array}{cc}a& s\\ 0& t\end{array}\right|}{\left|\begin{array}{cc}a& b\\ 0& d\end{array}\right|}=\frac{at}{ad}=\frac{t}{d} \end{gathered}$$

Step 8. Use of a ≠ 0 , b ≠ 0 , c ≠ 0 , d = 0

$\begin{array}{rcl}ax+by& =& s\\ cx& =& t or x=\frac{t}{c}\end{array}$

And,

$\begin{array}{rcl}a\frac{t}{c}+by& =& s\\ by& =& s-\frac{at}{c}\\ by& =& \frac{sc-at}{c}\\ y& =& \frac{sc-at}{bc}\end{array}$

Step 9. Finding x and y

By Cramer Rule,

$$\begin{gathered} x=\frac{\left|\begin{array}{cc}s& b\\ t& 0\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& 0\end{array}\right|}=\frac{-tb}{-bc}=\frac{t}{c} \\ y=\frac{\left|\begin{array}{cc}a& s\\ c& t\end{array}\right|}{\left|\begin{array}{cc}a& b\\ c& 0\end{array}\right|}=\frac{sc-at}{bc} \end{gathered}$$