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The slope of a line is a measure of its steepness, indicating how much the line rises or falls as you move along it. To find the slope between two points, such as (-2,-1) and (5,4), we can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] In our case, substituting the values gives:- Change in y: \(4 - (-1) = 5\)- Change in x: \(5 - (-2) = 7\) Thus, the slope \(m = \frac{5}{7}\). This tells us that for every 7 units you move to the right, the line moves up by 5 units.

An equation of a line can be represented in various forms, with one of the most common being the point-slope form. This form is useful when you know the slope and one point on the line.For the points (-2,-1) and (5,4) and the slope \( m = \frac{5}{7} \), the point-slope form is:\[ y - y_1 = m(x - x_1) \]Plugging in the values gets you:- From point (-2,-1), the line is: \[ y - (-1) = \frac{5}{7}(x - (-2)) \] Simplifying: \[ y + 1 = \frac{5}{7}(x + 2) \]To convert to standard form (Ax + By + C = 0), follow these steps:

• Multiply through by 7 to eliminate fractions: \( 7(y + 1) = 5(x + 2) \)
• Expand: \( 7y + 7 = 5x + 10 \)
• Rearrange to \( 5x - 7y + 3 = 0 \)

Now, the line equation is ready for further calculations, like finding distances.

To find the area of a triangle formed by points (2,3), (-2,-1), and (5,4), one effective method is Heron's Formula. Before using it, we compute the lengths of the sides using the Distance Formula.For three points on a plane, calculate the side lengths:

• Side a: Between (2,3) and (-2,-1), the length is \( 4\sqrt{2} \)
• Side b: Between (2,3) and (5,4), the length is \( \sqrt{10} \)
• Side c: Between (-2,-1) and (5,4), the length is \( \sqrt{74} \)

Next, determine the semi-perimeter \(s\) with:\[ s = \frac{a + b + c}{2} \]Then, Heron's formula to find the area \(A\) is:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]Calculating these values gives the area as approximately 4.94 square units. This approach is useful in cases where the vertex coordinates are known.

The distance from a point to a line in the coordinate plane can be precisely calculated using the point-to-line distance formula. This is particularly helpful to determine the shortest distance from a specific point to a defined line without drawing perpendicular lines manually.The formula to find this distance \(d\) is: \[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \] Where \(Ax + By + C = 0\) is the line equation, and \((x_0, y_0)\) is your point of interest.For the line equation \(5x - 7y + 3 = 0\) and point (2,3), substitute:

• A = 5, B = -7, C = 3
• Point: \(x_0 = 2, y_0 = 3\)

Calculating gives:\[ d = \frac{|5(2) + (-7)(3) + 3|}{\sqrt{5^2 + (-7)^2}} = \frac{1}{\sqrt{74}} \]This result reveals the shortest route between the point and the line, illustrating the usefulness of the Distance Formula.