Questlon 16 Which equation represents a line that passes through the points $$ (4,5) $$ and $$ (6,9) $$ ? $$ \begin{array}{ll}\text { A) } y=2 x-3 \ \text { (в) } y=3 x-7 \ \text { C) } y=\frac{1}{2} x+3 \ \text { D) } y=\frac{1}{3} x+7\end{array} $$

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To determine which equation represents the line passing through the points $$ (4,5) $$ and $$ (6,9) $$, follow these steps:

1. **Calculate the Slope (m):**
   
   The slope of a line passing through two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:
   $$
   m = \frac{y_2 - y_1}{x_2 - x_1}
   $$
   
   Applying the points $$ (4,5) $$ and $$ (6,9) $$:
   $$
   m = \frac{9 - 5}{6 - 4} = \frac{4}{2} = 2
   $$
   
2. **Use the Slope-Intercept Form:**
   
   The slope-intercept form of a line is:
   $$
   y = mx + b
   $$
   
   We already know $$ m = 2 $$. To find $$ b $$, plug in one of the points. Let's use $$ (4,5) $$:
   $$
   5 = 2(4) + b \implies 5 = 8 + b \implies b = -3
   $$
   
   Therefore, the equation of the line is:
   $$
   y = 2x - 3
   $$
   
3. **Verify with the Second Point:**
   
   Plugging $$ (6,9) $$ into $$ y = 2x - 3 $$:
   $$
   9 = 2(6) - 3 \implies 9 = 12 - 3 \implies 9 = 9 \quad \text{(True)}
   $$
   
4. **Compare with the Options:**

- **A)** $$ y = 2x - 3 $$ **✓**
   - B) $$ y = 3x - 7 $$
   - C) $$ y = \frac{1}{2}x + 3 $$
   - D) $$ y = \frac{1}{3}x + 7 $$

**Answer:**  
**A) $$ y = 2x - 3 $$**
The equation that represents the line passing through the points $$ (4,5) $$ and $$ (6,9) $$ is $$ y = 2x - 3 $$.

Questlon 16 Which equation represents a line that passes through the points \( (4,5) \) and \( (6,9) \) ? \[ \begin{array}{ll}\text { A) } y=2 x-3 \\ \text { (в) } y=3 x-7 \\ \text { C) } y=\frac{1}{2} x+3 \\ \text { D) } y=\frac{1}{3} x+7\end{array} \]

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