The value of lambda for which the lines 3x + | vedclass

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(B) The given lines are $3x + 4y = 5,$ $5x + 4y = 4,$ and $\lambda x + 4y = 6.$These lines are concurrent if the point of intersection of the first tw

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(B) The given lines are $3x + 4y = 5,$ $5x + 4y = 4,$ and $\lambda x + 4y = 6.$These lines are concurrent if the point of intersection of the first two lines lies on the third line.Subtracting the first equation from the second: $(5x + 4y) - (3x + 4y) = 4 - 5 \implies 2x = -1 \implies x = -\frac{1}{2}.$Substituting $x = -\frac{1}{2}$ into $3x + 4y = 5$: $3(-\frac{1}{2}) + 4y = 5 \implies -\frac{3}{2} + 4y = 5 \implies 4y = 5 + \frac{3}{2} = \frac{13}{2} \implies y = \frac{13}{8}.$The point of intersection is $(-\frac{1}{2}, \frac{13}{8}).$Since this point lies on $\lambda x + 4y = 6,$ we have $\lambda(-\frac{1}{2}) + 4(\frac{13}{8}) = 6.$$-\frac{\lambda}{2} + \frac{13}{2} = 6 \implies -\frac{\lambda}{2} = 6 - \frac{13}{2} = -\frac{1}{2} \implies \lambda = 1.$

The value of $\lambda$ for which the lines $3x + 4y = 5,$ $5x + 4y = 4,$ and $\lambda x + 4y = 6$ meet at a point is

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