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The **point-slope form** is a way to describe the equation of a straight line when you know a single point on the line and its slope. This form is written as:\[y - y_1 = m(x - x_1)\]Where:

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

This formula comes in handy, especially when you don't know the y-intercept but have other details.
Let's say you have a line passing through the point \((b, c)\) with a known slope \(d\).
By plugging these values into the point-slope equation, you get:\[y - c = d(x - b)\]This is an intermediate step in converting to other forms of linear equations. It helps build connections between the point, the slope, and the behavior of the line.

The **slope-intercept form** of a line's equation is one of the simplest and most popular ways to express a linear equation.This form is written as:\[y = mx + b\]Where:

• \( m \) is still the slope of the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

By knowing the slope and y-intercept, you can plot the entire line on a graph. For our purposes, you can rearrange a point-slope equation into this form if needed.
Starting from the point-slope equation, for example:\[y - c = d(x - b)\]Solve for \( y \), and you'll end up with the slope-intercept form. Even though this is not always necessary for getting to standard form, it's a useful stepping stone for visualizing the line quickly.

The **standard form** of a line's equation is another common format, different from the others in that it focuses on organizing the terms naturally and neatly:\[Ax + By = C\]Where:

• \( A \), \( B \), and \( C \) are integers.
• \( A \) is non-negative, and both \( A \) and \( B \) are not both zero.

Starting from the point-slope form of our exercise:\[y - c = d(x - b)\]We rearrange this into standard form by moving all terms involving \(x\) and \(y\) to one side, and constant terms to the other side. The results are equations like \(-dx + y = c - db\).
In cases where the coefficients aren't integers, multiply every term to eliminate fractions. You might prefer a positive \( A \), so multiplying the equation by \(-1\) might do the trick if needed. This ensures our answer isn't only correct but also "clean" and easy to interpret.