Question

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

(i) y = 5x $-$ 3 ; y = 4 $-$ 2x ; 2x + 3y = 8

Answer 1

Three equations will be concurrent if the solution of two equations satisfies the third equation.
We have:
y = 5x $-$ 3
⇒ 5x $-$ y = 3 -------------(1)
y = 4 $-$ 2x
⇒ 2x + y = 4 -------------(2)
2x + 3y = 8 -------------(3)
Adding (1) and (2), we get:
5x $-$ y = 3
2x + y = 4
-------------------
7x = 7
⇒ x = 1
Substituting the value of x in equation (1), we get:
5(1) $-$ y = 3
⇒ 5$-$ y = 3
⇒ y = 2
Here, x = 1 and y = 2 is the solution of equations (1) and (2).
Now, putting x = 1 and y = 2 in equation (3), we have:
LHS = 2x + 3y = $2\times 1+3\times 2=8$ = RHS
i.e., the solution of equations (1) and (2) satisfies equation (3).
∴ The given three equations are concurrent.