If 19xN = 2n - 5m, Find A) 7x1 B) 3 x 4/7 C) -4x3 D) 2/5 x 4/7

Here is the solution to the problem shown in the image.

: Given the binary operation: $M \times N = 2N - 5M$. We need to evaluate this operation for the following expressions:

a) Evaluate $7 \times 1$

Step 1: Identify the values for $M$ and $N$. For $7 \times 1$, we have $M = 7$ and $N = 1$.

Step 2: Substitute these values into the operation definition. $7 \times 1 = 2(1) - 5(7)$

Step 3: Perform the multiplications. $7 \times 1 = 2 - 35$

Step 4: Perform the subtraction. $7 \times 1 = \boxed{-33}$

b) Evaluate $3 \times \frac{4}{7}$

Step 1: Identify the values for $M$ and $N$. For $3 \times \frac{4}{7}$, we have $M = 3$ and $N = \frac{4}{7}$.

Step 2: Substitute these values into the operation definition. $3 \times \frac{4}{7} = 2\left(\frac{4}{7}\right) - 5(3)$

Step 3: Perform the multiplications. $3 \times \frac{4}{7} = \frac{8}{7} - 15$

Step 4: Convert to a common denominator and subtract. $3 \times \frac{4}{7} = \frac{8}{7} - \frac{15 \times 7}{7}$ $3 \times \frac{4}{7} = \frac{8}{7} - \frac{105}{7}$ $3 \times \frac{4}{7} = \frac{8 - 105}{7}$ $3 \times \frac{4}{7} = \boxed{-\frac{97}{7}}$

c) Evaluate $-4 \times 3$

Step 1: Identify the values for $M$ and $N$. For $-4 \times 3$, we have $M = -4$ and $N = 3$.

Step 2: Substitute these values into the operation definition. $-4 \times 3 = 2(3) - 5(-4)$

Step 3: Perform the multiplications. $-4 \times 3 = 6 - (-20)$ $-4 \times 3 = 6 + 20$

Step 4: Perform the addition. $-4 \times 3 = \boxed{26}$

d) Evaluate $\frac{2}{5} \times \frac{4}{7}$

Step 1: Identify the values for $M$ and $N$. For $\frac{2}{5} \times \frac{4}{7}$, we have $M = \frac{2}{5}$ and $N = \frac{4}{7}$.

Step 2: Substitute these values into the operation definition. $\frac{2}{5} \times \frac{4}{7} = 2\left(\frac{4}{7}\right) - 5\left(\frac{2}{5}\right)$

Step 3: Perform the multiplications. $\frac{2}{5} \times \frac{4}{7} = \frac{8}{7} - \frac{10}{5}$ $\frac{2}{5} \times \frac{4}{7} = \frac{8}{7} - 2$

Step 4: Convert to a common denominator and subtract. $\frac{2}{5} \times \frac{4}{7} = \frac{8}{7} - \frac{2 \times 7}{7}$ $\frac{2}{5} \times \frac{4}{7} = \frac{8}{7} - \frac{14}{7}$ $\frac{2}{5} \times \frac{4}{7} = \frac{8 - 14}{7}$ $\frac{2}{5} \times \frac{4}{7} = \boxed{-\frac{6}{7}}$