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When you're given two points on a graph and need to find the slope of the line that runs through them, you can use the slope formula. The slope formula is \ \(m = \frac{y_2 - y_1}{x_2 - x_1}\ \), where \(m\) represents the slope, and \((x_1, y_1)\) and \((x_2, y_2)\) are your given points. This formula helps you determine how steep the line is.
For example, in the given exercise, we have points \((-2, 4)\) and \((3, -1)\). By plugging these values into the slope formula, we get \ \(m = \frac{-1 - 4}{3 - (-2)} = \frac{-5}{5} = -1\ \).
So, the slope here is \ -1\, indicating a line that declines from left to right.

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to investigate geometric shapes and properties. In coordinate geometry, every point is defined by a pair of numbers typically written as \((x, y)\).
Identifying the coordinates of points is fundamental because every calculation and concept in this domain revolves around these x and y values. Think of a graph with a horizontal x-axis and a vertical y-axis.
The point \((-2, 4)\) means you're at \(-2\) on the x-axis and \(4\) on the y-axis, so you move left on the x-axis and then up on the y-axis. Similarly, the point \((3, -1)\) takes you right 3 units and down 1 unit.
By plotting these points, you can accurately describe lines, angles, distances, and more.

A linear equation represents a straight line on a graph. The general form of a linear equation in two variables (x and y) is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. The y-intercept is where the line crosses the y-axis.
For instance, if the equation of the line is \(y = -1x + b\) and you want to find \(b\), you can use any of the points given. Using point \((-2, 4)\), substitute x and y into the equation:
\(4 = -1(-2) + b\). Simplifying gives \(4 = 2 + b\), so \(b = 2\). Therefore, your equation is \(y = -1x + 6\).
This line has a slope of \ -1\, matching our findings. Understanding linear equations helps in predicting how changing one variable affects another, a key concept in algebra and coordinate geometry.