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Problem Overview:

We are given two points in a plane:

• $A(-4, -1)$
• $B(5, 0)$

We need to find the coordinates of the points $M$ and $N$ such that:
(a) $3\overrightarrow{AM} + 2\overrightarrow{BM} = \overrightarrow{0}$
(b) $-3\overrightarrow{BN} + 2\overrightarrow{AB} = 2\overrightarrow{NA}$

Step 1: Solve Part (a)

We express $\overrightarrow{AM}$ and $\overrightarrow{BM}$ in terms of coordinates.

Vectors:

• $\overrightarrow{AM} = (x - (-4), y - (-1)) = (x + 4, y + 1)$
• $\overrightarrow{BM} = (x - 5, y - 0) = (x - 5, y)$

Substitute these into the equation $3\overrightarrow{AM} + 2\overrightarrow{BM} = \overrightarrow{0}$:
$3(x + 4, y + 1) + 2(x - 5, y) = (0, 0)$

Expand the equation:
$(3(x + 4) + 2(x - 5), 3(y + 1) + 2y) = (0, 0)$

Simplify each component:

1. For the $x$-component:
   $3(x + 4) + 2(x - 5) = 3x + 12 + 2x - 10 = 5x + 2 = 0$
   Solve:
   $5x = -2 \implies x = -\frac{2}{5}$

2. For the $y$-component:
   $3(y + 1) + 2y = 3y + 3 + 2y = 5y + 3 = 0$
   Solve:
   $5y = -3 \implies y = -\frac{3}{5}$

Thus, the coordinates of $M$ are:
$M\left(-\frac{2}{5}, -\frac{3}{5}\right)$

Step 2: Solve Part (b)

Vectors:

• $\overrightarrow{BN} = (x - 5, y - 0) = (x - 5, y)$
• $\overrightarrow{AB} = (5 - (-4), 0 - (-1)) = (9, 1)$
• $\overrightarrow{NA} = ((-4) - x, (-1) - y) = (-4 - x, -1 - y)$

Substitute these into the equation $-3\overrightarrow{BN} + 2\overrightarrow{AB} = 2\overrightarrow{NA}$:
$-3(x - 5, y) + 2(9, 1) = 2(-4 - x, -1 - y)$

Expand each term:

1. $-3\overrightarrow{BN} = (-3(x - 5), -3y) = (-3x + 15, -3y)$
2. $2\overrightarrow{AB} = (2 \cdot 9, 2 \cdot 1) = (18, 2)$
3. $2\overrightarrow{NA} = (2(-4 - x), 2(-1 - y)) = (-8 - 2x, -2 - 2y)$

Substitute back into the equation:
$(-3x + 15 + 18, -3y + 2) = (-8 - 2x, -2 - 2y)$

Simplify each component:

1. For the $x$-component:
   $-3x + 33 = -8 - 2x$
   Solve:
   $33 + 8 = 3x - 2x \implies x = 41$

2. For the $y$-component:
   $-3y + 2 = -2 - 2y$
   Solve:
   $2 + 2 = 3y - 2y \implies y = 4$

Thus, the coordinates of $N$ are:
$N(41, 4)$

Final Answer:

• $M\left(-\frac{2}{5}, -\frac{3}{5}\right)$
• $N(41, 4)$

Would you like further explanations for any step?

Related Questions:

1. How do we interpret vectors in coordinate geometry problems like this?
2. Why do we need to balance both the $x$- and $y$-components in vector equations?
3. Can we solve this problem using a different approach, such as matrices?
4. What are the geometrical meanings of $M$ and $N$ here?
5. How can we verify the solutions for $M$ and $N$ are correct?

Tip:

Always carefully define the components of vectors when solving equations, as clarity helps avoid errors in sign or calculation.