Find an equation of the line passing through the pair of points. \[ (4,-1),(-16,14) \] Sketch the line.

To find the equation of the line passing through the points \((4, -1)\) and \((-16, 14)\), we will follow these steps: 1. **Calculate the slope (m)** of the line using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1) = (4, -1)\) and \((x_2, y_2) = (-16, 14)\). 2. **Use the point-slope form** of the equation of a line: \[ y - y_1 = m(x - x_1) \] We can use either of the two points for \((x_1, y_1)\). 3. **Convert the equation to slope-intercept form** \(y = mx + b\) if necessary. Let's start with the calculations. ### Step 1: Calculate the slope (m) Using the points: - \(x_1 = 4\), \(y_1 = -1\) - \(x_2 = -16\), \(y_2 = 14\) Calculating the slope: \[ m = \frac{14 - (-1)}{-16 - 4} = \frac{14 + 1}{-16 - 4} = \frac{15}{-20} = -\frac{3}{4} \] ### Step 2: Use the point-slope form Using the point \((4, -1)\): \[ y - (-1) = -\frac{3}{4}(x - 4) \] This simplifies to: \[ y + 1 = -\frac{3}{4}(x - 4) \] ### Step 3: Convert to slope-intercept form Distributing the slope: \[ y + 1 = -\frac{3}{4}x + 3 \] Subtracting 1 from both sides: \[ y = -\frac{3}{4}x + 2 \] Now we have the equation of the line: \[ y = -\frac{3}{4}x + 2 \] ### Sketching the Line To sketch the line, we can plot the two points \((4, -1)\) and \((-16, 14)\) and draw a line through them. Let's summarize the results and sketch the line. 1. **Equation of the line**: \[ y = -\frac{3}{4}x + 2 \] 2. **Points to plot**: - \((4, -1)\) - \((-16, 14)\) Now, I will sketch the line based on these points. ### Sketching the Line The line will have a negative slope, indicating it will go downwards from left to right. The y-intercept is at \(2\) (where the line crosses the y-axis). Let's proceed to sketch the line. (Note: As a text-based AI, I cannot create visual sketches directly, but you can plot the points and draw the line on graph paper or using graphing software.)