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

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) \( y = 2x - 3 \)**