91Ó°ÊÓ

The point-slope form of a linear equation is a tool used to find the equation of a line when you know one point on the line and the slope. It is expressed as:

• \( y - y_1 = m(x - x_1) \)

Here, \( (x_1, y_1) \) represents a known point on the line, and \( m \) is the slope. This form is particularly handy because it directly connects the slope to a specific point, making it easy to plug in numbers and solve.
The process involves:

• Identifying the slope, which in this case is 0 for a horizontal line.
• Choosing the given point, which is \((1, 3)\).

Substitute these values into the point-slope formula. It transforms the equation into a practical form that can further be simplified or converted into other forms, such as the standard form.

The standard form of a line is one of the common ways to express linear equations. Its formula is:

• \( Ax + By = C \)

In this equation:

• \( A, B, \text{ and } C \) are integers.
• \( A \geq 0 \).

The main advantage of the standard form is its versatility in representing vertical and horizontal lines clearly. For horizontal lines, like the line \( y = 3 \), the standard form is simply \(0x + 1y = 3\).
Converting from another form, such as slope-intercept

• Start with the simplified equation \( y = 3 \).
• Recognize this already follows the \( Ax + By = C \) pattern with \( B = 1 \) and \( C = 3 \).

This flexibility in conversion makes the standard form a reliable choice in various problem scenarios.

The slope of a line is a measure of its steepness and direction. It's calculated as the ratio of the change in the vertical coordinates to the change in the horizontal coordinates between two distinct points:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

In this exercise, however, the line is parallel to a horizontal line.
For any horizontal line, the slope (

• \( m \)) is 0
• because the vertical change is zero.

Whenever a line is parallel to a horizontal line, its slope remains 0, indicating no incline. This property simplifies many calculations, as it means the line is parallel to the x-axis, leading directly to equations like \( y = C \) without any x-component, indicating constant output regardless of x. This simplicity helps maintain focus on problem-solving without complex computations.