Q. 61 Complete the proof of Cramer鈥檚... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 61 (page 741)

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

Short Answer

Expert verified

The Cramer Rule is Proved.

Step by step solution

Step 1. Given Information

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

We will solve this task by assuming

(a) $a = 0, b 鈮� 0, c 鈮� 0, d 鈮� 0$

(b) $a 鈮� 0, b = 0, c 鈮� 0, d 鈮� 0$

(d) $a 鈮� 0, b 鈮� 0, c 鈮� 0, d = 0$

We change into the system of equations $a x + b y = s c x + d y = t$

Step 2. Use of a=0,b≠0,c≠0,d≠0

$b y = s o r y = \frac{s}{b} - - - - - (1) c x + d y = t - - - - (2)$

From (1) and (2)

$\begin{array}{rcl} c x + d \frac{s}{b} & = & t \\ c x & = & t - \frac{d s}{b} = \frac{b t - d s}{b} \\ x & = & \frac{b t - d s}{b c} \end{array}$

Step 3. Finding x and y

Using Cramer's formula

$\begin{array}{rcl} x & = & \frac{|\begin{matrix} s & b \\ t & d \end{matrix}|}{|\begin{matrix} 0 & b \\ c & d \end{matrix}|} \\ = & \frac{s d - t b}{- b c} \\ = & \frac{t b - s d}{b c} \end{array}$

And,

$\begin{array}{rcl} y & = & \frac{|\begin{matrix} 0 & s \\ c & t \end{matrix}|}{|\begin{matrix} 0 & b \\ c & d \end{matrix}|} \\ = & \frac{- s c}{- b c} \\ = & \frac{s}{b} \end{array}$ '

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

$\begin{array}{rcl} a x & = & s o r x = \frac{s}{a} \\ c x + d y & = & t \end{array}$

And,

\begin{array}{rcl} c \frac{s}{a} + d y & = & t \\ d y & = & t - \frac{c s}{a} = \frac{t a - c s}{a} \\ y & = & \frac{a t - c s}{a d} \end{array}

Step 5. Finding x and y

$x = \frac{|\begin{matrix} s & 0 \\ t & d \end{matrix}|}{|\begin{matrix} a & 0 \\ c & d \end{matrix}|} = \frac{s d}{a d} = \frac{s}{a} y = \frac{|\begin{matrix} a & s \\ c & t \end{matrix}|}{|\begin{matrix} a & 0 \\ c & d \end{matrix}|} = \frac{a t - c s}{a d}$

Step 6. Use of a≠0,b≠0,c=0,d≠0

$a x + b y = s d y = t o r y = \frac{t}{d}$

$a x + b \frac{t}{d} = s a x = s - \frac{b t}{d} = \frac{s d - b t}{d} x = \frac{s d - b t}{a d}$

Step 7. Finding x and y

$x = \frac{|\begin{matrix} s & b \\ t & d \end{matrix}|}{|\begin{matrix} a & b \\ 0 & d \end{matrix}|} = \frac{s d - t b}{a d} y = \frac{|\begin{matrix} a & s \\ 0 & t \end{matrix}|}{|\begin{matrix} a & b \\ 0 & d \end{matrix}|} = \frac{a t}{a d} = \frac{t}{d}$

Step 8. Use of a≠0,b≠0,c≠0,d=0

$\begin{array}{rcl} a x + b y & = & s \\ c x & = & t o r x = \frac{t}{c} \end{array}$

And,

\begin{array}{rcl} a \frac{t}{c} + b y & = & s \\ b y & = & s - \frac{a t}{c} \\ b y & = & \frac{s c - a t}{c} \\ y & = & \frac{s c - a t}{b c} \end{array}

Step 9. Finding x and y

By Cramer Rule,

$x = \frac{|\begin{matrix} s & b \\ t & 0 \end{matrix}|}{|\begin{matrix} a & b \\ c & 0 \end{matrix}|} = \frac{- t b}{- b c} = \frac{t}{c} y = \frac{|\begin{matrix} a & s \\ c & t \end{matrix}|}{|\begin{matrix} a & b \\ c & 0 \end{matrix}|} = \frac{s c - a t}{b c}$

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91影视

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

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Use of a=0,b≠0,c≠0,d≠0

Step 3. Finding x and y

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

Step 5. Finding x and y

Step 6. Use of a≠0,b≠0,c=0,d≠0

Step 7. Finding x and y

Step 8. Use of a≠0,b≠0,c≠0,d=0

Step 9. Finding x and y

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