Question

Without drawing the graphs, show that the following equations are of concurrent lines :

(ii) 2x + y = 6 ; x + 2y = 6 ; 7x $-$ 4y = 6

Answer 1

Three equation will be concurrent if the solution of two equations satisfies the third equation.
We have:
2x + y = 6 ...(1)
x + 2y = 6 ...(2)
7x $-$ 4y = 6 ...(3)
Multiplying equation (1) by 2, we get:
4x + 2y = 12 ...(4)
Subtracting (2) from (4), we get:
4x + 2y = 12
x + 2y = 6
$-$ $-$ $-$
-------------------
3x = 6
⇒ x = 2
Substituting the value of x in equation (1), we get:
2(2) + y = 6
⇒ 4 + y = 6
⇒ y = 2
Here, x = 2 and y = 2 is the solution of equations (1) and (2).
Now, putting x = 2 and y = 2 in equation (3), we have:
LHS = 7x $-$ 4y = $7\times 2-4\times 2=6$ = RHS
i.e., the solution of equations (1) and (2) satisfies equation (3).
∴ The given three equations are concurrent.