Intersecting the Y-Axis: A Comprehensive Guide to Determining the Point of Intersection

Understanding how to graph and analyze linear equations is a fundamental concept in algebra. One of the key points to consider when graphing a linear equation is the point where the line intersects the y-axis. This article will explore the process of finding the y-intercept of a given line and provide a detailed example to make the concept clear.

Introduction to Linear Equations and the Y-Axis

A linear equation in two variables, such as 5x 3y 15, represents a straight line in a coordinate plane. The y-axis is the vertical line where the x-coordinate is always 0. The point where the line crosses this axis is known as the y-intercept.

Identifying the Y-Intercept

To find the y-intercept, you need to determine the value of y when x is 0. This is because at the y-axis, the x-coordinate is 0, and the y-coordinate is the value of y where the line crosses the y-axis.

Solving for the Y-Intercept

Given the equation of a line: 5x 3y 15.

Substitute x 0 into the equation. This simplifies the equation to: 5(0) 3y 15. Perform the multiplication: 0 3y 15 or simply 3y 15. Solve for y by dividing both sides of the equation by 3: y 15 / 3 5.

So, the y-intercept of the line is the point where x 0 and y 5. This point is (0, 5).

The Significance of the Y-Intercept

The y-intercept is a critical point in the graph of a linear equation and holds several important implications:

Starting Point: The y-intercept represents the starting point of the line when it crosses the y-axis. It provides the value of the dependent variable (y) when the independent variable (x) is at its baseline (0). Parameters: In real-world applications, the y-intercept often represents the initial value or starting condition of a system. For example, if the equation represents a cost or revenue model, the y-intercept might indicate the fixed cost or revenue at zero production or sales. Comparison and Interpretation: Knowing the y-intercept allows for a better understanding of the linear function. It helps in making predictions and analyzing trends.

Graphing the Line

To graph the line 5x 3y 15, follow these steps:

Determine the y-intercept (0, 5). Solve for another point by choosing a value for x, other than 0, and finding the corresponding y-value. For instance, if x 3, substitute into the equation: 5(3) 3y 15 which simplifies to 15 3y 15 or 3y 0. Solving for y gives y 0. So, another point is (3, 0). PLOT these points on a coordinate plane and draw a straight line through them.

Conclusion

Identifying the point where a line intersects the y-axis is a basic yet crucial skill in algebra. By substituting x 0 into the equation, you can easily determine the y-intercept. This point (0, 5) in the given example helps in graphing the line accurately and provides essential information about the behavior of the linear function.

Related Topics

Linear Equations: General forms and properties of linear equations. Graphing Linear Equations: Techniques for plotting lines on a coordinate plane. Y Intercept Calculator: Online tools for finding y-intercepts of various linear equations.

Understanding how to find the y-intercept is a stepping stone to mastering more complex algebraic concepts. Practice with different equations and explore real-world applications to deepen your understanding and proficiency.