find the gradient of the line \( (-3,6),(6,1) \) using graph

To find the **gradient** (also known as the **slope**) of the line passing through the points \((-3, 6)\) and \( (6, 1) \), you can use the following method: ### Step 1: Understand the Gradient Formula The gradient \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] ### Step 2: Plug in the Given Points Given the points: - \((x_1, y_1) = (-3, 6)\) - \((x_2, y_2) = (6, 1)\) Substitute these values into the formula: \[ m = \frac{1 - 6}{6 - (-3)} = \frac{-5}{9} \] ### Step 3: Interpret the Result - **Gradient \( m = -\frac{5}{9} \)** This means that for every 9 units you move to the right (positive direction on the x-axis), the line descends by 5 units (negative direction on the y-axis). ### Visual Representation If you were to graph these points and draw the line between them: 1. **Plot the Points:** - Point A: \((-3, 6)\) - Point B: \( (6, 1) \) 2. **Draw the Line:** - Connect the two points with a straight line. 3. **Determine the Gradient:** - From Point A to Point B, move **9 units right** (from \(x = -3\) to \(x = 6\)). - Move **5 units down** (from \(y = 6\) to \(y = 1\)). The slope \( m = \frac{\text{Vertical Change}}{\text{Horizontal Change}} = \frac{-5}{9} \) confirms the gradient. ### Conclusion The **gradient of the line** passing through the points \((-3, 6)\) and \( (6, 1)\) is: \[ m = -\frac{5}{9} \]