91Ó°ÊÓ

The slope-intercept form is a fundamental concept in linear equations. It provides a straightforward way to understand the relationship between the slope and the y-intercept of a line. The formula is represented as \( y = mx + b \),
where \( m \) represents the slope of the line, and \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This form is particularly useful for quickly grasping the overall direction and position of a line on a coordinate plane.

- **Slope (\( m \))**: It indicates how steep the line is. A positive slope means the line ascends from left to right, while a negative slope means it descends.
- **Y-Intercept (\( b \))**: This is where the line intersects the y-axis. It's the value of \( y \) when \( x = 0 \).
In our exercise, the slope \( m \) is given as \(-\frac{4}{3}\). By finding the corresponding y-intercept, using a known point on the line, we can express the equation of the line in simple, clear terms that describe its movement across the coordinate plane.

Point-slope form is an essential tool when you know a point on a line and the slope of that line but don't yet have the full equation. The formula is:
\( y - y_1 = m(x - x_1) \)
Here, \((x_1, y_1)\) represents a point on the line, and \( m \) is the slope. This form is perfect for constructing the equation when these specific details are available.

- **Usage**: Ideal for situations where the slope and one point are known because it directly incorporates these values into the equation.
- **Conversion**: It can be easily converted into slope-intercept form by solving for \( y \).
In our exercise, we use the point-slope formula with the point \((-5, 0)\) and slope \(-\frac{4}{3}\). As a result, the equation becomes \( y - 0 = -\frac{4}{3}(x + 5) \),
which simplifies to the slope-intercept form \( y = -\frac{4}{3}x - \frac{20}{3} \). This showcases how we can derive an equation from known values.

Standard form offers another way to express a line's equation, which can be advantageous in certain scenarios like solving systems of equations or finding intercepts easily. The standard form is given by:
\( Ax + By = C \)
where \( A \), \( B \), and \( C \) are integers, and \( A \) should be a non-negative integer.

- **No Fractions**: Ideally, the coefficients \( A \), \( B \), and \( C \) are integers, which often requires multiplying the entire equation to clear any fractions.
- **Advantages**: This form is particularly useful for analyzing vertical and horizontal intercepts.

In our exercise, having derived the slope-intercept form \( y = -\frac{4}{3}x - \frac{20}{3} \), we convert it by eliminating the fractions (multiplying through by 3) to get \( 3y = -4x - 20 \).
Then, rearrange to form \( 4x + 3y = -20 \), a clean representation of the line in standard form. This process highlights how different forms of the same equation can serve various computational needs.