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Have you ever pondered over the enigmatic world of linear equations? What secrets do they hold when graphed on a coordinate grid? Today, we're about to embark on an intriguing journey to uncover the truth behind one such mystery.
Imagine you're presented with a coordinate grid, and on it, two lines are sketched – a blue one and a brown one. The blue line represents the equation y = x - 7, and the brown line symbolizes y = 12x - 1. Now, here's the twist: what happens when we plug in x = 8 into these equations? What value does y take on for each line?
As we dive into the first equation, y = x - 7, we find that when x equals 8, y = 8 - 7, which simplifies to y = 1. We observe this point plotted on the blue line, marked as (8, 1). But what about the brown line? Does it yield the same result?
Curiosity piqued, we turn our attention to the brown line's equation, y = 12x - 1. Plugging in x = 8, we calculate y = 12 * 8 - 1, resulting in y = 95. However, something unexpected occurs – the point (8, 5) is not on the brown line, but rather, it lies on a completely different trajectory.
Now, here's a question that beckons: why is this significant? The answer lies in the nature of solutions to linear equations. If x = 8 were a solution to both equations simultaneously, the lines would intersect at that point. But they don't. The blue and brown lines are distinct, with no intersection at (8, 1) or (8, 5). This discrepancy tells us that x = 8 is not a solution to both equations.
But wait, there's more! Visual confirmation further solidifies our findings. When we graph these equations, we notice two distinct points on the grid – one for each line. They do not overlap, reinforcing the fact that x = 8 is not a solution to both equations.
In conclusion, we've unraveled the mystery of these linear equations on the coordinate grid. The blue line, y = x - 7, and the brown line, y = 12x - 1, do not intersect at x = 8, confirming that this value is not a solution to both equations. So, next time you encounter a coordinate grid with linear equations, remember to question, compute, and verify before drawing any conclusions.
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