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Have you ever wondered how a simple line on a graph can reveal so much about its nature? Today, we're going to unlock the mystery behind one of the most fundamental properties of a line: the y-intercept. Let's embark on this algebraic journey together and uncover what it truly signifies.
What if I asked you to identify the y-intercept of the line shown before you? Take a moment to pause and ponder this question. What would you say?
The y-intercept is not just a random point on the graph; it's where the line intersects the y-axis. But how do we find it exactly? This is where algebra steps in. You might recall the equation of a line, which is y = mx + b. Here, 'm' represents the slope, and 'b' is the y-intercept we're trying to find.
First things first, we need to determine the slope of the line. Remember, slope is the change in y over the change in x. By examining the given points, we can calculate this change. From the point (4,0) to (7,-2), the change in y is -2, and the change in x is 3. Thus, the slope is -2/3.
Now, let's pause and consider this: what if we used a different pair of points? Would we get the same slope? Let's verify this by using the coordinates (-2,4) and (7,-2). The change in y is -6, and the change in x is 9, which once again gives us a slope of -2/3. Consistency is key, and we've confirmed our initial observation.
With the slope in hand, we can now rewrite the line's equation as y = -2/3x + b. But how do we find 'b', the elusive y-intercept? We substitute one of the given points into the equation and solve for 'b'. Let's choose the point (-2,4) for simplicity.
By substituting x = -2 and y = 4 into our equation, we get 4 = -2/3(-2) + b. Simplifying this, we find that b = 4 - 4/3. Through some basic arithmetic, we discover that b equals 8/3, or 2 and 2/3.
And there it is, the y-intercept we were searching for all along. It's at the point 2 and 2/3 on the y-axis. This journey through algebraic concepts has brought us full circle, from identifying the slope to uncovering the y-intercept. As we conclude, let's reflect on the initial question: what is the y-intercept of this line? Our exploration has provided us with the answer, and the beauty of mathematics has once again proven itself.
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