For what value of $\lambda$, the equations $7x-5y+3=0$, $2x+3y=8$ and $4x-6y+\lambda =0$ are consistent?

A system of three linear equations in two variables is consistent if there exists a point $(x,y)$ that satisfies all three equations.First, solve the first two equations:$7x-5y=-3$$2x+3y=8$ Multiply (1) by $3$ and (2) by $5$:$21x-15y=-9$$10x+15y=40$ Adding them: $31x=31⟹x=1$.Substitute $x=1$ in (2): $2(1)+3y=8⟹3y=6⟹y=2$.For the third equation $4x-6y+\lambda =0$ to be consistent, the point $(1,2)$ must satisfy it:$4(1)-6(2)+\lambda =0$$4-12+\lambda =0$$-8+\lambda =0⟹\lambda =8$.