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When dealing with linear equations, the point-slope form is a useful method for writing the equation of a line. This form is particularly handy when you know the slope of a line and a point through which the line passes. The point-slope formula is given by

• \[ y - y_1 = m(x - x_1) \]

where

• \( m \) is the slope,
• \( (x_1, y_1) \) is the point on the line.

To use this formula, insert the values of the slope and coordinates of the given point. For the example given, with a slope of \( \frac{3}{2} \) and the point (5, -6), plugging these values into the formula yields:

• \[ y + 6 = \frac{3}{2}(x - 5) \].

From this structure, it's straightforward to manipulate the equation into different forms, such as the standard form.

The standard form of a linear equation is a conventional way of writing equations, making them simple to interpret and compare. It is expressed as

• \( Ax + By = C \),

where A, B, and C are integers. In solving the exercise, after finding the initial equation in the point-slope form, converting it to standard form involves a series of straightforward steps:

• First, simplify any fractions present in the equation, usually by multiplying the entire equation by the denominator of the fraction involved.


• Rearrange the equation so that all the variables are on one side and the constant is on the other.

For instance, multiplying the given equation \( y + 6 = \frac{3}{2}x - \frac{15}{2} \) by 2 eliminates fractions, leading to:

• \[ 2y + 12 = 3x - 15 \].

Finally, moving all terms to one side of the equation results in the standard form:

• \[ 3x - 2y = 27 \].

The slope of a line is a measure of its steepness and direction. Conceptually, it tells how much the line rises for each unit it runs horizontally. The formula for slope, often denoted as \( m \), is

• \[ m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} \].

In the exercise, the slope is given as \( \frac{3}{2} \), indicating that for every 2 units the line moves horizontally, it rises by 3 units. Understanding slope is crucial for graphing lines and for the creation of equations in point-slope and other forms. When you plug the slope into the point-slope form with a specific point, it allows for the construction of the line's complete equation.

Coordinate geometry, also known as analytic geometry, involves graphing and solving equations on a coordinate plane, a system with an x-axis and y-axis. It allows algebra and geometry to intersect perfectly and is foundational to understanding linear equations.

• Provides a visual representation of equations and inequalities.


• Makes it easier to determine the intersection, distance, and slope of lines.

In our exercise, using coordinate geometry principles helps visualize how a line with a specific slope passes through a given point. By identifying the coordinates (in this case, (5, -6)) and applying it to equations like point-slope, we solidify our understanding of how numbers translate into shapes on a graph. This approach supports critical learning, helping you interpret abstract algebraic formulations by connecting them to geometric figures on a graph. By practicing with both the equation and its graphical representation, the concept of lines and how they operate within the coordinate plane becomes clearer.