Question

Find an equation for the line with the given properties. Express your answer using either the general form or the slope-intercept form of the equation of a line, whichever you prefer. Perpendicular to the line \(y=2 x-3 ;\) containing the point (1,-2)

Short answer

The equation is \(y = -\frac{1}{2}x - \frac{3}{2}\).

Step-by-step solution

- Identify the given line's slope

The given line's equation is in slope-intercept form (\(y = mx + b\)), where \(m\) is the slope. For the line \(y = 2x - 3\), the slope \(m\) is 2.

- Determine the slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. The slope of the given line is 2, so the slope of the perpendicular line will be \(-\frac{1}{2}\).

- Use the point-slope formula

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\), where \(m\) is the slope, and \((x_1, y_1)\) is a point on the line. Here, the point is \((1, -2)\), and the slope is \(-\frac{1}{2}\).

- Substitute the point and the slope into the point-slope formula

Plugging in the given point (1, -2) and the slope \(-\frac{1}{2}\) into the point-slope formula, we get: \(y - (-2) = -\frac{1}{2}(x - 1)\).

- Simplify the equation

Simplify the equation to get it into slope-intercept form (\(y = mx + b\)). First, simplify the left side: \(y + 2 = -\frac{1}{2}x + \frac{1}{2}\). Then, isolate y: \(y = -\frac{1}{2}x + \frac{1}{2} - 2\). Simplify further: \(y = -\frac{1}{2}x - \frac{3}{2}\).

Key concepts

slope

The term 'slope' in geometry describes the steepness or incline of a line. It is usually denoted by the letter 'm'. The slope is calculated as the 'rise' over the 'run', or the change in y-values divided by the change in x-values between two points on a line.
In mathematical terms, for two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \((m)\) is given by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This shows how much the line goes up or down for each unit it moves horizontally. A positive slope means the line ascends as it moves from left to right, while a negative slope means it descends.

slope-intercept form

The slope-intercept form of a line's equation is one of the most common ways to express a linear equation. It is written as \({ y = mx + b }\), where:

• \