Problem 3 - Precalculus: Mathematics for Calculus

Question

The point-slope form of the equation of the line with slope 3 passing through the point \((1,2)\) is __________.

Step-by-Step Solution

Answer

The point-slope form is \(y - 2 = 3(x - 1)\).

Step 1: Understand the Point-Slope Form Equation

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

Step 2: Identify Given Values

In this problem, we are given the slope \( m = 3 \) and the point \((1, 2)\) through which the line passes. So, \(x_1 = 1\) and \(y_1 = 2\).

Step 3: Substitute Into the Formula

Substitute the given slope and point into the point-slope form equation: \(y - 2 = 3(x - 1)\).

Step 4: Verify the Equation

Check that each part is correctly substituted: \(y - y_1 = 3(x - 1)\) should have \(y_1\) as 2 and \(x_1\) as 1, confirming \(y - 2 = 3(x - 1)\) is correct.

Key Concepts

Slope of a Line

In mathematics, the slope of a line is a measure of its steepness, usually described as the ratio of the vertical change to the horizontal change between two points on the line. This is often referred to as "rise over run."
For any two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) can be calculated using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

A positive slope indicates that as we move from left to right, the line rises. Conversely, a negative slope means the line falls as we move from left to right. If the slope is zero, the line is horizontal, and if it's undefined, the line is vertical.
In the context of the point-slope form of a linear equation, the slope is crucial for determining the direction and angle at which the line extends. It is denoted by \( m \), and it directly affects the equation \( y - y_1 = m(x - x_1) \).

Linear Equations

Linear equations are equations of the first degree, meaning they graph as straight lines in a coordinate plane. A linear equation can take several forms:

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

The point-slope form is particularly useful when you know a single point on the line and the slope, as it directly incorporates these pieces of information into the equation format.
This form simplifies deriving equations of lines when specific points and slope values are known. For example, if a line has a slope \( m = 3 \) and passes through the point \((1, 2)\), we can express its equation as \( y - 2 = 3(x - 1) \).
Linear equations are fundamental in algebra as they model relationships and solve real-life problems where rates of change are constant. They appear frequently in mathematical modelling and problem-solving scenarios.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, amalgamates algebra and geometry to study geometric figures using a coordinate system. It allows us to describe geometric shapes and analyze properties using algebraic equations.
In coordinate geometry, the entire plane is composed of ordered pairs of numbers \( (x, y) \) that denote each point's position. Each point corresponds to an intersection in the plane, specified by its \( x \)-coordinate and \( y \)-coordinate in the Cartesian system.
With this system, we can explore various geometric concepts, such as:

• The slope of lines, describing how steep or flat they are
• Equations of lines, circles, and other curves
• Distance and midpoint calculations between points

The point-slope form is a vital tool in coordinate geometry as it directly uses a point \( (x_1, y_1) \) on the line and the slope \( m \) to define the line's equation, allowing for simpler computation and visualization of lines in the plane. This interplay of algebra and geometry deepens our understanding of spatial relationships and is crucial in fields such as engineering, physics, and computer graphics.