what is the equation of the line graphed below

Guri Bee

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23-10-2024

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Unlocking the Secrets of Linear Equations: Finding the Equation from a Graph

Understanding the relationship between a line and its equation is a fundamental concept in algebra. This article will guide you through the process of determining the equation of a line directly from its graph. We'll explore the key concepts and illustrate them with practical examples.

What is the Equation of a Line?

The equation of a line is a mathematical representation that describes the relationship between the x and y coordinates of all the points on that line. It can be written in various forms, but the most common one is the slope-intercept form:

y = mx + c

Where:

• y represents the vertical coordinate (dependent variable)
• x represents the horizontal coordinate (independent variable)
• m is the slope of the line, which measures its steepness and direction
• c is the y-intercept, the point where the line crosses the y-axis.

Steps to Find the Equation of a Line from its Graph:

1. Identify the y-intercept: Look for the point where the line intersects the y-axis. The y-coordinate of this point is the y-intercept (c).

2. Calculate the slope: Choose any two distinct points on the line. Let's call them (x1, y1) and (x2, y2). The slope (m) is calculated using the formula:

   m = (y2 - y1) / (x2 - x1)

3. Substitute the values of m and c into the slope-intercept form: Now that you have the slope (m) and y-intercept (c), plug them into the equation y = mx + c.

Example:

Let's say we have a line that crosses the y-axis at (0, 2) and passes through the point (4, 6).

1. y-intercept (c) = 2

2. Slope (m): Using the points (0, 2) and (4, 6)

   m = (6 - 2) / (4 - 0) = 4 / 4 = 1

3. Equation of the line: Substituting m = 1 and c = 2 into the equation:

   y = 1x + 2 or simply y = x + 2

Additional Tips:

• If the line is horizontal, its slope is 0 (m = 0), and the equation will be of the form y = c, where c is the y-intercept.
• If the line is vertical, its slope is undefined. It cannot be represented in slope-intercept form. Instead, the equation will be of the form x = a, where a is the x-intercept.

Conclusion:

By understanding the slope-intercept form and the relationship between a line and its equation, you can easily determine the equation of any line from its graph. This skill is crucial in various applications, including physics, engineering, and economics, where understanding the behavior of lines is essential.