Question

Explain why you should find the slope \(m\) first when finding the equation of a line passing through two points.

Short answer

Finding the slope first when finding the equation of a line passing through two points is critical because the slope gives us information about the rate at which the line rises or falls. This information can then be used to find the y-intercept and hence the equation of the line.

Step-by-step solution

Understanding the concept of slope

Before finding the equation of the line passing through two points, it is important to understand the concept of a slope. The term 'slope' in mathematics is used to describe the measure of the steepness of a line. In a two-dimensional coordinate system, slope (\(m\)) is represented as: \(m = (y_2 - y_1) / (x_2 - x_1)\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points on the line.

Importance of the Slope

The slope of a line provides two main pieces of information. The first is the direction in which the line rises or falls and the second is how much it rises or falls. Therefore, without calculating the slope first, it is impossible to determine the exact equation of a line.

Finding the equation of a line using the slope

The equation of a line can be found by first calculating the slope using the formula stated above. This slope can then be substituted into the general form of the line equation, \(y = mx + b\), to solve for the y-intercept (\(b\)). As, it's known that \((x_1, y_1)\) is a point on the line, hence, we can substitute these values into \(y = mx + b\) to find the value of the y-intercept.

Key concepts

Slope of a Line

Understanding the slope of a line is a foundational concept in algebra and geometry. The slope is a measure of the line's steepness and direction. In mathematical terms, the slope is calculated as the ratio of the change in the y-coordinate (\text{rise}) to the change in the x-coordinate (\text{run}). Mathematically, it's expressed as:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of any two points on the line. If you were walking along the line, the slope tells you how much you'd have to climb or descend between points.

Why Find the Slope First?

When plotting a line, the slope is an essential characteristic because it defines the angle of the line relative to the horizontal axis. To draw a line on a graph, you'd require a starting point and a direction - the slope provides that direction. Knowing the slope lets you predict where the line will go next, regardless of its location in the coordinate system. Additionally, when comparing two lines, their slopes can tell us if they are parallel, perpendicular, or neither.

Coordinate System

The coordinate system, also known as the Cartesian coordinate system, is crucial for graphing equations and understanding geometric relations on a plane. This system is composed of two perpendicular lines, generally referred to as axes. The horizontal axis is the x-axis, and the vertical one is the y-axis. The point where both axes intersect is called the origin, typically labeled as \( (0, 0) \).

Plotting Points and Lines

To plot a point in this system, you use an ordered pair \( (x, y) \), where \( x \) corresponds to the horizontal location and \( y \) to the vertical location. When plotting a line, you need at least two points to determine its position. The relationship between the points, given by the slope, dictates how the line will be positioned in the system. The slope tells us if the line rises or falls as it moves from left to right, and it's this concept that enables us to properly represent lines and even curves on a graph.

Y-Intercept

The y-intercept of a line is simply where the line crosses the y-axis. In other words, it's the y-coordinate of the point where the line intersects the y-axis, which occurs when the x-coordinate is zero. The y-intercept is commonly denoted as \( b \) in the slope-intercept form of a line's equation, which is written as:
\[ y = mx + b \]
Here, \( m \) stands for the slope of the line. When you plot the line using the slope and y-intercept, you start at \( b \), the y-intercept, and then follow the slope to find another point on the line.

Importance in the Line Equation

Knowing the y-intercept is essential for graphically representing the line and for understanding how the line relates to the y-axis. It can be seen as the starting point of the line in the y-direction. Having both the slope, indicating the direction and steepness, and the y-intercept, indicating the starting vertical position, makes it possible to draw the entire line across the coordinate system.