Question

A point on a line and its slope are given. Find the point-slope form of the equation of the line. $$ P=(2,4) ; m=-\frac{3}{4} $$

Short answer

The point-slope form is \( y - 4 = -\frac{3}{4}(x - 2) \).

Step-by-step solution

Identify the point and the slope

The given point is \(P=(2,4)\) and the slope is \(m=-\frac{3}{4}\).

Recall the point-slope form

The point-slope form of a line is given by \(y - y_1 = m(x - x_1)\) where \( (x_1, y_1) \) is a point on the line and \(m\) is the slope.

Substitute the given values

Plug in \(x_1 = 2\), \(y_1 = 4\), and \(m = -\frac{3}{4}\) into the point-slope formula: \[ y - 4 = -\frac{3}{4}(x - 2) \]

Simplify the equation (Optional)

Optionally, you can distribute the slope and simplify: \[ y - 4 = -\frac{3}{4}x + \frac{3}{2} \]

Key concepts

Point-Slope Form

The point-slope form of a linear equation is useful when you know a point on the line and the slope of the line. This form is written as: \( y - y_1 = m(x - x_1) \). Here, \( (x_1, y_1) \) is the given point and \( m \) is the slope. This form makes it easy to graph a line quickly or convert it to other forms, like the slope-intercept form.

Slope

The slope \( m \) measures the steepness of a line and is calculated as the 'rise' over 'run' or the change in \( y \) over the change in \( x \). In our example, the slope \( m = -\frac{3}{4}\) which means:

• The line drops 3 units for every 4 units it moves to the right
• A negative slope indicates the line is going downhill from left to right
• A positive slope would mean the line is going uphill

Understanding the slope helps us visualize the line's direction and steepness and is crucial for forming equations.

Linear Equations

Linear equations represent straight lines on a coordinate plane. They can be expressed in multiple forms:

• Standard Form: \( Ax + By = C \)
• Slope-Intercept Form: \( y = mx + b \)
• Point-Slope Form: \( y - y_1 = m(x - x_1) \)

Each form has its advantages. Point-slope is great for when we have a point and the slope. Knowing how to convert between different forms is an essential skill for solving various problems related to linear equations.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, involves studying geometry using the Cartesian coordinate system. Key elements include points, lines, and curves:

• A point is represented by an ordered pair \( (x, y) \)
• A line is often represented by a linear equation \( y = mx + b \) or \( Ax + By = C \)

In our example, the point \( P=(2, 4) \) and slope \( m = -\frac{3}{4} \) define a unique line. Understanding these concepts is fundamental to solving problems and analyzing geometric shapes within a coordinate plane.