The given points $(4, 3)$ and $(-1, 6)$ can be used to calculate several properties and values, such as the distance between the points, the slope of the line passing through them, or the equation of the line. Here's how each is determined:

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1. Distance Between the Points

The formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substituting the coordinates $(4, 3)$ and $(-1, 6)$:

$d = \sqrt{((-1) - 4)^2 + (6 - 3)^2}$ $d = \sqrt{(-5)^2 + (3)^2}$ $d = \sqrt{25 + 9} = \sqrt{34}$

So, the distance between the points is $\sqrt{34} \approx 5.83$.

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2. Slope of the Line

The formula for the slope of the line passing through two points is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the coordinates:

$m = \frac{6 - 3}{-1 - 4} = \frac{3}{-5} = -\frac{3}{5}$

The slope of the line is $-\frac{3}{5}$.

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3. Equation of the Line

The slope-intercept form of a line is:

$y = mx + b$

Using the slope $m = -\frac{3}{5}$ and one of the points, say $(4, 3)$, to find $b$:

$3 = -\frac{3}{5}(4) + b$ $3 = -\frac{12}{5} + b$ $b = 3 + \frac{12}{5} = \frac{15}{5} + \frac{12}{5} = \frac{27}{5}$

Thus, the equation of the line is:

$y = -\frac{3}{5}x + \frac{27}{5}$

Or equivalently, in standard form:

$3x + 5y = 27$

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Summary of Results:

1. Distance: $\sqrt{34} \approx 5.83$
2. Slope: $-\frac{3}{5}$
3. Equation of the Line: $y = -\frac{3}{5}x + \frac{27}{5}$ or $3x + 5y = 27$

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Would you like to explore any of these properties in more detail, or solve related problems?

5 Related Questions:

1. What is the midpoint of the line segment joining these two points?
2. How do we determine if another point lies on the same line?
3. What is the perpendicular distance from the origin to this line?
4. Can the equation of the line be represented parametrically?
5. How do you find the equation of a line perpendicular to this one passing through a given point?

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Tip:

When calculating slopes, double-check signs to ensure accuracy, especially when subtracting negative values!