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The slope of a line, often denoted by \( m \), is a critical concept in coordinate geometry. It measures the steepness or incline of a line. To calculate the slope, you need two distinct points on the line. The formula for slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Here, \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the points. The subtraction \( y_2 - y_1 \) gives the difference in the y-values. Similarly, \( x_2 - x_1 \) gives the difference in the x-values. This ratio indicates how much the y-value changes for a unit change in x.

In the exercise, the points given are \((-3,6)\) and \((1,2)\). Substituting into the formula, we find:

• \( y_2 - y_1 = 2 - 6 = -4 \)
• \( x_2 - x_1 = 1 - (-3) = 4 \)

Thus, the slope \( m \) is \( \frac{-4}{4} = -1 \), meaning the line decreases one unit in y for each increase of one unit in x.

Once the slope is determined, you can use the point-slope form to write the equation of the line. This form is very handy when you have one point on the line and the slope. The equation is: \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) are the coordinates of the known point, and \( m \) is the slope. This form allows you to directly substitute the point and slope values to create an initial version of the line's equation.

For the exercise, using the point \((-3,6)\) and the slope of \(-1\), the equation becomes:

• \( y - 6 = -1(x + 3) \)

This equation represents the line in point-slope form, making it easy to visualize how the y-value changes with the x-value.

The linear equation of a line is a way to describe the line with a general formula. In its simplest form, the equation is \( y = mx + b \), often called the slope-intercept form. Here, \( m \) stands for slope, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. Once you have used point-slope form to establish the initial equation, you usually convert it to the slope-intercept form by algebraically rearranging it.

In the given problem, simplifying \( y - 6 = -1(x + 3) \) gives:

• Distribute \(-1\): \( y = -x - 3 + 6 \)
• Simplify: \( y = -x + 3 \)

This linear equation makes it straightforward to find any point on the line by choosing a value for \( x \) and calculating \( y \).

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. By plotting points on a Cartesian plane, any geometric figure can be represented algebraically, allowing for an analytical approach to understand geometry. The coordinate plane has two axes, the x-axis (horizontal) and y-axis (vertical), which intersect at the origin (0,0).

In solving problems like finding the equation of a line:

• Identify and plot points on the graph, such as \((-3,6)\) and \((1,2)\).
• Use these coordinates with algebraic formulas to find relationships, like the slope or intercept.
• Visualize geometric concepts like lines as algebraic entities \((y = -x + 3)\), which helps in understanding their properties and behavior.

Coordinate geometry combines drawing with mathematics, making it a powerful tool for solving geometric problems.