A binary operation $\Delta$ is defined on the set of real numbers $R$ by $a \Delta b = a + b + 4ab$. Given that $(5 \Delta k) = (4 \Delta k) + 9$, we need to find the value of $k$.

1. Problem Description

A binary operation $\Delta$ is defined on the set of real numbers $R$ by $a \Delta b = a + b + 4ab$. Given that $(5 \Delta k) = (4 \Delta k) + 9$, we need to find the value of $k$.

2. Solution Steps

First, we use the definition of the binary operation $\Delta$ to express $5 \Delta k$ and $4 \Delta k$.

$a \Delta b = a + b + 4ab$

$5 \Delta k = 5 + k + 4(5)(k) = 5 + k + 20k = 5 + 21k$

$4 \Delta k = 4 + k + 4(4)(k) = 4 + k + 16k = 4 + 17k$

Now we substitute these expressions into the given equation:

$(5 \Delta k) = (4 \Delta k) + 9$

$5 + 21k = 4 + 17k + 9$

$5 + 21k = 13 + 17k$

Now, we solve for $k$.

$21k - 17k = 13 - 5$

$4k = 8$

$k = \frac{8}{4}$

$k = 2$

3. Final Answer

The value of $k$ is 2.