3. Find the Cartesian equation of the line that passes through the points $$ A $$ and $$ B $$. a) $$ A(-3,6,4), B(9,7,1) $$ b) $$ A(6,4,-5), B(2,-4,3) $$ c) $$ A(3,5,-5), B(7,3,4) $$ 4. Write the Cartesian equation of the line through the points a) $$ (5,-4) $$ and $$ (7,3) $$ b) $$ (2,-4) $$ and $$ (7,2) $$

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**3. Cartesian Equation of a Line in 3D**

**a) For $$ A(-3,6,4) $$ and $$ B(9,7,1) $$:**

1. Compute the direction vector:
   $$
   \vec{d} = B - A = (9 - (-3),\; 7 - 6,\; 1 - 4) = (12,\; 1,\; -3)
   $$
2. Write the symmetric equations of the line:
   $$
   \frac{x - (-3)}{12} = \frac{y - 6}{1} = \frac{z - 4}{-3}
   $$
   which simplifies to:
   $$
   \frac{x + 3}{12} = \frac{y - 6}{1} = \frac{z - 4}{-3}
   $$

---

**b) For $$ A(6,4,-5) $$ and $$ B(2,-4,3) $$:**

1. Compute the direction vector:
   $$
   \vec{d} = B - A = (2 - 6,\; -4 - 4,\; 3 - (-5)) = (-4,\; -8,\; 8)
   $$
2. Simplify the direction vector by dividing by $$-4$$:
   $$
   \vec{d} = (-4,-8,8) = -4(1,2,-2) \quad \Longrightarrow \quad (1,2,-2)
   $$
   (Alternatively, you can use $$(-1,-2,2)$$; the line remains the same.)
3. Write the symmetric equations using the original (or simplified) direction vector. Using the original form:
   $$
   \frac{x - 6}{-4} = \frac{y - 4}{-8} = \frac{z + 5}{8}
   $$
   Or, using the simplified vector $$(-1,-2,2)$$ with $$ A(6,4,-5) $$:
   $$
   \frac{x - 6}{-1} = \frac{y - 4}{-2} = \frac{z + 5}{2}
   $$

---

**c) For $$ A(3,5,-5) $$ and $$ B(7,3,4) $$:**

1. Compute the direction vector:
   $$
   \vec{d} = B - A = (7 - 3,\; 3 - 5,\; 4 - (-5)) = (4,\; -2,\; 9)
   $$
2. Write the symmetric equations of the line:
   $$
   \frac{x - 3}{4} = \frac{y - 5}{-2} = \frac{z + 5}{9}
   $$

---

**4. Cartesian Equation of a Line in 2D**

**a) For points $$ (5,-4) $$ and $$ (7,3) $$:**

1. Compute the slope:
   $$
   m = \frac{3 - (-4)}{7 - 5} = \frac{7}{2}
   $$
2. Use the point-slope form with point $$ (5,-4) $$:
   $$
   y + 4 = \frac{7}{2}(x - 5)
   $$
3. To write in standard form, multiply both sides by 2:
   $$
   2y + 8 = 7x - 35
   $$
4. Rearranging:
   $$
   7x - 2y - 43 = 0
   $$

---

**b) For points $$ (2,-4) $$ and $$ (7,2) $$:**

1. Compute the slope:
   $$
   m = \frac{2 - (-4)}{7 - 2} = \frac{6}{5}
   $$
2. Use the point-slope form with point $$ (2,-4) $$:
   $$
   y + 4 = \frac{6}{5}(x - 2)
   $$
3. Multiply both sides by 5:
   $$
   5(y + 4) = 6(x - 2)
   $$
4. Expand and simplify:
   $$
   5y + 20 = 6x - 12
   $$
5. Rearranging:
   $$
   6x - 5y - 32 = 0
   $$
**3. Cartesian Equation of a Line in 3D** **a) For $$ A(-3,6,4) $$ and $$ B(9,7,1) $$:** $$ \frac{x + 3}{12} = \frac{y - 6}{1} = \frac{z - 4}{-3} $$ **b) For $$ A(6,4,-5) $$ and $$ B(2,-4,3) $$:** $$ \frac{x - 6}{-4} = \frac{y - 4}{-8} = \frac{z + 5}{8} $$ **c) For $$ A(3,5,-5) $$ and $$ B(7,3,4) $$:** $$ \frac{x - 3}{4} = \frac{y - 5}{-2} = \frac{z + 5}{9} $$ --- **4. Cartesian Equation of a Line in 2D** **a) For points $$ (5,-4) $$ and $$ (7,3) $$:** $$ 7x - 2y - 43 = 0 $$ **b) For points $$ (2,-4) $$ and $$ (7,2) $$:** $$ 6x - 5y - 32 = 0 $$

Find the Cartesian equation of the line that passes through the points and .
a)
b)
c)

Write the Cartesian equation of the line through the points
a) and
b) and

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