Questions: Which of the following lines is perpendicular to the line passing through (1,-1) and (2,1) ? y-2 x=-5 y+2 x=8 2 y+x=3 2 y-x=5 y+3 x=-4

Solution Steps

To determine which line is perpendicular to the line passing through the points \((1, -1)\) and \((2, 1)\), we need to:

1. Calculate the slope of the line passing through the given points.
2. Determine the negative reciprocal of this slope, as the slope of a perpendicular line is the negative reciprocal.
3. Check which of the given lines has this slope.

To determine which line is perpendicular to the line passing through the points \((1, -1)\) and \((2, 1)\), we first need to find the slope of this line.

The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the given points \((1, -1)\) and \((2, 1)\): \[ m = \frac{1 - (-1)}{2 - 1} = \frac{1 + 1}{2 - 1} = \frac{2}{1} = 2 \]

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, if the slope of the original line is \(2\), the slope of the perpendicular line is: \[ m_{\perp} = -\frac{1}{2} \]

We need to find which of the given lines has a slope of \(-\frac{1}{2}\). We will convert each equation to the slope-intercept form \(y = mx + b\) to identify the slope \(m\).

1. \(y - 2x = -5\) \[ y = 2x - 5 \quad \text{(slope is 2)} \]

2. \(y + 2x = 8\) \[ y = -2x + 8 \quad \text{(slope is -2)} \]

3. \(2y + x = 3\) \[ 2y = -x + 3 \implies y = -\frac{1}{2}x + \frac{3}{2} \quad \text{(slope is -}\frac{1}{2}\text{)} \]

4. \(2y - x = 5\) \[ 2y = x + 5 \implies y = \frac{1}{2}x + \frac{5}{2} \quad \text{(slope is }\frac{1}{2}\text{)} \]

5. \(y + 3x = -4\) \[ y = -3x - 4 \quad \text{(slope is -3)} \]

Final Answer