a+b=1,2b+c=1 and a+2c=1

Question

system of equation solve it.

Answer 1

Elimination method

Given equations are a + b = 1 ---> (1)

2 b + c = 1  ---> (2)

a + 2 c = 1 -------> (3) 

multiply each side by - 1 in eq (1) 

We get the equation - a - b = - 1 -------> (4)

To eliminate the a value add the equations (4)&(3).

- a - b = - 1

a + 2 c = 1

______________

- b + 2 c = 0 -------> (5)

multiply each side by 2 in eq (5) 

We get the equation - 2 b +4 c = 0 --------> (6)

To eliminate the b value add the equations (2)&(6).

2 b + c = 1

- 2 b + 4 c = 0

_______________

 5 c = 1 

c = 1 / 5

Substitute the c value in eq (4)

- b + 2 c = 0

- b + 2( 1/ 5) = 0

- b = - 2 / 5

b = 2 / 5.

Substitute the  b  value in eq (1) 

a + b = 1

a + 2 / 5 = 1

a = 1 - 2 / 5

a = 3 / 5.

Solution : a = 3 / 5 , b = 2 / 5 and  c = 1 / 5.

Answer 2

Substitution method :

The equations are

a + b = 1   → (1)

2b + c = 1 → (2)

a + 2c = 1 → (3)

Solve equation (3) for a, since the y has a coefficient of 1.

a + 2c = 1 → a = 1 - 2c

Solve equation (2) for b.

2b + c = 1 → b = (1 - c )/2

Substitute the values a = 1 - 2c and b = (1 - c )/2 in equation (1) and solve for c.

1 - 2c + (1 - c )/2 = 1

2c = (1 - c )/2

4c = 1 - c → 5c = 1 → c = 1/5

⇒ c = 1/5.

Substitute the value c = 1/5 in b = 2c and solve for b.

b = 2(1/5)

⇒ b = 2/5.

Substitute the value b = 2/5 in equation (1) and solve for a.

a + (2/5) = 1 → a = 1 - 2/5 = 3/5.

⇒ a = 3/5.

Solution of the system is a = 3/5, b = 2/5, and c = 1/5.