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Understanding the point-slope form of a line is essential for writing equations that represent a line in coordinate geometry.

The point-slope formula \(y - y_1 = m(x - x_1)\) offers a direct method to write an equation of a line when you know the slope \(m\) and a point \( (x_1, y_1) \) on the line. Here's the formula at a glance:

• \((y_1)\) is the y-coordinate of the known point.
• \((x_1)\) is the x-coordinate of the same point.
• \((m)\) represents the slope of the line.

\((y - y_1\)) and \((x - x_1\)) show the distance from any point \((x, y)\) on the line to the known point. The equation intuitively indicates how far up or down you need to go \( (y - y_1)\) and how far left or right \( (x - x_1)\) to stay on the line, as guided by the slope, \(m\).

For the given problem, the line goes through the point \( (10, 6) \) and has a slope of 7. Substituting these values into the point-slope form, we get the equation of the line as it relates to its slope and a specific point on the line. The ease of using the point-slope form lies in its straightforward substitution process, making it a favorite for quickly drafting the framework of a linear equation.

The slope of a line is a measure that indicates how steep the line is and its direction.

Mathematically, the slope \(m\) is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two distinct points on a line: \[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \].

• If the slope is positive, the line rises from left to right.
• If the slope is negative, it falls from left to right.
• If the slope is zero, the line is horizontal.
• If the slope is undefined, the line is vertical.

In our exercise, the given slope is 7, which means for every one unit the line moves horizontally to the right (positive x-direction), it rises by 7 units in the vertical direction (positive y-direction). This characteristic of the slope defines the inclination of the line and is key to building the equation that represents such a line. The procedure of using slope values can also be applied to find the equation of parallel or perpendicular lines, as they will have slopes that are either equal or negative reciprocals, respectively.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used to describe lines in the coordinate plane.

These equations can be represented in various forms, including:

• Standard form: \(Ax + By = C\), with \(A, B\), and \(C\) as integers, and \(A\) should be positive.
• Slope-intercept form: \(y = mx + b\), where \(m\) is the slope, and \(b\) is the y-intercept.
• Point-slope form: \(y - y_1 = m(x - x_1)\) as discussed earlier.

The standard form is particularity useful for analyzing equations quickly, identifying intercepts and is required in certain academic contexts. Converting to standard form, as done in Step 3 of our exercise, involves moving all terms involving variables to one side of the equation and the constant to the other side, while also keeping the leading coefficient positive. In our provided exercise, rearranging the point-slope form equation resulted in the standard form \(7x - y = 64\), which expresses the relationship between \(x\) and \(y\) in an equivalent, yet different, manner than the point-slope form, showcasing the versatility of linear equations in mathematics.