Points Available: 5 Question 1: A straight line passes through \( (1,5) \) and has a gradient of 2 . The equation of the line is: A \( y=2 x-3 \)

To determine the correct equation of the straight line that passes through the point \( (1, 5) \) with a gradient (slope) of 2, let's analyze the given option and derive the correct equation. ### Given: - **Point**: \( (1, 5) \) - **Gradient (Slope)**: \( m = 2 \) ### Option A: \[ y = 2x - 3 \] #### Checking Option A: Let's verify if the point \( (1, 5) \) lies on the line \( y = 2x - 3 \): 1. Substitute \( x = 1 \) into the equation: \[ y = 2(1) - 3 \] \[ y = 2 - 3 \] \[ y = -1 \] 2. The calculated \( y \)-value is \(-1\), but the point has \( y = 5 \). Therefore, the point \( (1, 5) \) **does not** lie on the line \( y = 2x - 3 \). **Conclusion**: **Option A is incorrect.** ### Deriving the Correct Equation: To find the correct equation of the line, we'll use the **point-slope form** of a linear equation: \[ y - y_1 = m(x - x_1) \] Where: - \( m \) is the slope. - \( (x_1, y_1) \) is the given point. 1. Plug in the known values: \[ y - 5 = 2(x - 1) \] 2. Distribute and simplify: \[ y - 5 = 2x - 2 \] \[ y = 2x - 2 + 5 \] \[ y = 2x + 3 \] ### **Correct Equation**: \[ \boxed{y = 2x + 3} \] This line has a gradient of 2 and correctly passes through the point \( (1, 5) \).