Examine the consistency of the system of equations

Examine the consistency of the system of equations in Exercises 1 to 6.

1. x + 2y = 2 and 2x + 3y = 3 

|A|= 3- 4

|A|= -1

|A|≠ 0

Equations has unique solutions. Hence equations are consistence

 

 2. 2x – y = 5 and x + y = 4

|A|= 2+1

|A|= 3

|A|≠ 0

Equations has unique solutions. Hence equations are consistence

3. x + 3y = 5 and 2x + 6y = 8

|A|= 6-6

|A|= 0

Hence equations are inconsistence

4. x + y + z = 1 , 2x + 3y + 2z = 2 and ax + ay + 2az = 4

Solution

Equation third is divided by a we get

x + y + z = 1

2x + 3y + 2z = 2

x + y + 2z = 4/a

Hence |A| ≠0

So Equations has unique solutions. Hence equations are consistence

 5. 3x–y – 2z = 2, 2y – z =-1 and  –3x – 5y = 3

Multiply both matrices we get

Hence it is inconsistence

6. 5x – y + 4z = 5,2x + 3y + 5z = 2 and 5x – 2y + 6z = –1

|A|≠0

Hence equations are consistence