The Intersection of Two Lines: Solving the System of Equations

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Have you ever wondered how two lines interact on a coordinate grid? What happens when they meet? In this article, we'll explore a fascinating system of linear equations and discover the point where these lines intersect.

A Visual Clue to the Solution

Imagine a coordinate grid with two lines graphed – one in blue and the other in brown. The blue line represents the equation ( y = -2x - 2 ), while the brown line represents ( y = -\frac{1}{4}x + 5 ). Take a moment to visually inspect the grid. Can you guess where these lines meet? What is the X and Y pair that satisfies both equations?

A Point of Intersection

Upon closer inspection, it appears that the lines intersect at a specific point. It looks like the point (-4, 6) is where both lines cross. But is this point truly the solution to our system of equations? Let's verify.

Verifying the Solution

To confirm that (-4, 6) is indeed the intersection point, we'll substitute X = -4 into both equations and see if we get Y = 6.

For the Blue Line:

Using the equation ( y = -2x - 2 ): [ y = -2(-4) - 2 ] [ y = 8 - 2 ] [ y = 6 ]

The blue line confirms that when X = -4, Y = 6.

For the Brown Line:

Using the equation ( y = -\frac{1}{4}x + 5 ): [ y = -\frac{1}{4}(-4) + 5 ] [ y = 1 + 5 ] [ y = 6 ]

The brown line also confirms that when X = -4, Y = 6.

Conclusion: The Intersection Point

After verification, we can confidently say that the point (-4, 6) is where the blue and brown lines intersect. This point satisfies both equations, making it the solution to our system of linear equations.

So, the next time you see a system of equations, take a moment to visualize the lines and their intersection. It might just lead you to a fascinating discovery.

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