Find the value of n for which the following equations are consistent

Question

x - 2y = 3n
2x + 4y = n
x + 3y = -2

Answer 1

x - 2y = 3n
2x + 4y = n
intersect at (7/4 n, -5/8 n)

2x + 4y = n
x + 3y = -2
intersect at ( (3n+8)/2 , -(n+4)/2 )

To make the intersections the same point, we need n = 16

Answer 2

OR

Multiply second equation by 3.

6 x + 12 y = 3 n

First equation is:

x - 2 y = 3 n

3 n = 3 n

6 x + 12 y = x - 2 y

Subtract x to both sides.

5 x + 12 y = - 2 y

Subtract 12 y to both sudes.

5 x = - 14 y

- 14 y = 5 x

y = - 5 / 14 x

Replace y with - 5 / 14 in the third equation.

x + 3 y = - 2

x + 3 • ( - 5 / 14 x ) = - 2

x - 15 / 14 x = - 2

14 / 14 x - 15 / 14 x = - 2

- x / 14 = - 2

Multiply both sides by - 1

x / 14 = 2

Multiply both sides by 14

x = 28

Put this value in equation:

y = - 5 / 14 x

y = - 5 • 28 / 14 = - 140 / 14

y = - 10

Put x = 28 and y = - 10 in second equation.

2 x + 4 y = n

2 • 28 + 4 • ( - 10 ) = n

56 - 40 = n

n = 16