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Chapter 1: Q49E (page 22)

a. Find all solutionsx1,x2,x3,x4 of the system .

x2=12(x1+x3),x3=12(x2+x4)

b. In partrole="math" localid="1659677484607" (a) , is there a solution with x1=1andx4=13 ?

Short Answer

Expert verified

a. For arbitrary x3and x4, x1=3x3鈭�2x4,x2=2x3鈭�x4

b. Yes, x1=1,x2=5,x3=9,x4=13

Step by step solution

01

(a)Step 1: Consider the equations to find other equations

Find the expression for,x1.

x2=12(x1+x3)2x2=x1+x3x1=2x2鈭�x3鈥夆赌夆赌夆赌夆赌夆赌夆赌�......(1)

Find the expression for, x2.

x3=12(x2+x4)2x3=x2+x4x2=2x3鈭�x4鈥夆赌夆赌夆赌夆赌夆赌夆赌夆�夆�夆�夆��......(2)

02

Express the variable in terms of other variables

Re-write x1in terms of x3andx4 .

x1=2x2鈭�x3x1=2(2x3鈭�x4)鈭�x3=3x3鈭�2x4鈥夆赌夆赌夆赌夆赌夆赌夆赌�......(3)

Therefore, x1=3x3鈭�2x4,x2=2x3鈭�x4

03

(b)Step 3: Using the above equations, find the values of the variables

Consider the given values of the variables.

x1=1andx4=13

Substitute these values in the above equations to find the values of the remaining variables.

Consider the equation (3).

x1=3x3鈭�2x41=3x3鈭�2脳13x3=273=9

Consider the equation (2).

x2=2x3鈭�x4=2脳9鈭�13=18鈭�13=5

Therefore, x1=1,x2=5,x3=9,x4=13.

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