91Ó°ÊÓ

The slope-intercept form of a linear equation is a widely used way to express an equation of a line. It is given by the formula:
\[ y = mx + b \]
Where:

• \( y \) is the dependent variable or the output of the function.
• \( x \) is the independent variable or the input of the function.
• \( m \) represents the slope of the line, indicating how steep the line is.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

This form is particularly useful when you need to quickly identify the slope and the y-intercept of the line. For example, consider the line equation given as:
\[ y = -\frac{1}{2}x + 2 \]
Here, the slope \( m \) is -\frac{1}{2}, which means the line goes downwards as you move from left to right. The y-intercept is 2, indicating that the line crosses the y-axis at the point (0,2). This form is very convenient for easily graphing lines and understanding how changes in \( x \) affect \( y \).

Standard form of a linear equation is an alternative way of expressing the equation of a line. It is represented as:
\[ Ax + By + C = 0 \]
Where:

• \( A, B, \) and \( C \) are integers, and \( A \) should not be negative.
• \( x, y \) are variables.

This form is particularly convenient when dealing with systems of equations or when you need to rearrange terms. In our exercise, the line equation was given as:
\[ x + 2y - 4 = 0 \]
To convert this into standard form while maintaining the parallel feature, we adjust for any specific point. If we want a line through (2,-3) parallel to this, we will end up with the equation:
\[ x + 2y + 4 = 0 \]
In this exercise, the process involved rearranging the equation and maintaining integer values for the coefficients, which simplifies the analysis and manipulation of algebraic expressions.

The point-slope form is another common way to write the equation of a line. It is useful when you know one point on the line and the slope. The formula is:
\[ y - y_1 = m(x - x_1) \]
Where:

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

This form is especially helpful for writing the equation of a line when you have a single point and the line's slope. In the exercise, the point-slope form was utilized to derive the equation of a line by substituting the known point \((2, -3)\) and the slope \(-\frac{1}{2}\):
\[ y + 3 = -\frac{1}{2}(x - 2) \]
After performing some algebraic manipulation to simplify the expression, it was eventually converted to other forms. Deploying the point-slope formula gives a very direct way to generate a line equation, especially when aiming to express the line's behavior relative to its slope and a particular point, before transforming it into simpler forms like slope-intercept or standard form.