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The Point-Slope Form of a linear equation is a powerful tool for writing the equation of a line when you know the slope and a point on the line. It is generally expressed as:

• \[ y - y_1 = m(x - x_1) \]

In this equation, \( (x_1, y_1) \) represents a known point on the line, and \( m \) is the slope of the line.

To apply the point-slope form, you just substitute the values you know for \( m \), \( x_1 \), and \( y_1 \). In our exercise, we have a point \((-1, 3)\) and a slope of \(2\). So, we plug these into the formula to get:

• \[ y - 3 = 2(x - (-1)) \]

This representation in point-slope form makes it easy to convert it to other forms like the slope-intercept form by simplifying algebraically.

The slope of a line is a measure of its steepness, typically represented by the letter \( m \). It is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two distinct points on the line:

• \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. In our example, the points

• \((-2, -3)\) and \((2, 5)\)

are used to calculate the slope.

Plug in the values:

• \[ m = \frac{5 - (-3)}{2 - (-2)} = \frac{8}{4} = 2 \]

This means the line increases by 2 units on the y-axis for every 1 unit it moves to the right along the x-axis.

The uniform calculation of the slope ensures accuracy and is essential in identifying relationships between lines, such as parallel lines.

Lines are parallel if they run next to each other without ever intersecting, having the same slope but different y-intercepts. This characteristic is key when creating new lines parallel to existing ones.

When a line is parallel to a given line, it implies they share the same slope. In our exercise, the original line you refer to passes through the points

• \((-2, -3)\) and \((2, 5)\)

, with a calculated slope of \( m = 2 \).

If you want to find the equation of a new line parallel to this with a different y-intercept, you maintain the slope \( m = 2 \) and use a new point to determine the equation. From our point-slope form, the parallel line through the point \((-1, 3)\) becomes:

• \[ y = 2x + 5 \]

In essence, parallel lines have this repetitive quality in their slope, reflecting their inability to meet and showing up consistently in various scenarios with parallelism in mathematical problems.