Understanding Intercepts of a Linear Equation

Linear equations are fundamental in algebra and serve as the building blocks for more advanced mathematical concepts. In this article, we will explore the concept of intercepts in the context of the equation Y 5/2x - 2, specifically focusing on finding both the x-intercept and the y-intercept, and how to graph the equation.

Intercepts: A Fundamental Concept

In the Cartesian coordinate system, intercepts are the points at which a graph intersects the x-axis and the y-axis. Two types of intercepts are particularly important: the x-intercept and the y-intercept.

The Y-Intercept

The y-intercept of a line is the point at which the graph crosses the y-axis. This occurs when x0. To find the y-intercept, we follow these steps:

Substitute x with 0 in the equation. Simplify the equation. Identify the y-coordinate of the point.

Let's apply these steps to the given equation, Y 5/2x - 2: Substitute x0 into the equation: Y 5/2(0) - 2 Simplify: Y 0 - 2 -2 The y-intercept is the point (0, -2).

To verify this, we can plot the point (0, -2) on the y-axis of a graph.

The X-Intercept

The x-intercept of a line is the point at which the graph crosses the x-axis. This occurs when y0. To find the x-intercept, we follow these steps:

Set Y to 0. Solve for x. Identify the x-coordinate of the point.

Let's find the x-intercept for the given equation, Y 5/2x - 2: Set Y to 0: 0 5/2x - 2 add 2 to both sides: 2 5/2x multiply both sides by 2/5: x 4/5 The x-intercept is the point (4/5, 0).

Verification: By substituting x 4/5 into the equation, we get Y 5/2(4/5) - 2 2 - 2 0, which confirms that the x-intercept is indeed (4/5, 0).

Graphing the Equation

To graph the equation Y 5/2x - 2, we can plot the intercepts and draw a straight line through them.

First, mark the y-intercept (0, -2) on the y-axis and the x-intercept (4/5, 0) on the x-axis. Then, draw a straight line connecting these two points. This line represents the graph of the equation.

Additional Tips for Graphing

While the intercepts provide a good starting point for graphing, you can also use additional points for accuracy. Here’s a step-by-step guide:

Pick an x-value and solve for the corresponding y-value. Pick another x-value and solve for the corresponding y-value. Plot the points and draw a line.

For example, let's choose x 1 and x -1: When x 1, Y 5/2(1) - 2 5/2 - 2 1/2, so the point is (1, 1/2). When x -1, Y 5/2(-1) - 2 -5/2 - 2 -9/2, so the point is (-1, -9/2).

Plot these points and draw a straight line connecting all the points to trace the graph of the equation.