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The point-slope formula is a powerful tool for writing the equation of a line when you know a point on the line and the slope. The formula itself is expressed as: \( y - y_1 = m(x - x_1) \). Here:

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

This setup allows you to write a linear equation quickly by plugging in one point and the slope. Once you substitute these values, you can find the equation that describes your line. If you're starting with an exercise like ours, you might need to first find the slope using other available information, such as another point on the line. Then, simply plug the known point coordinates and the slope into the formula. This is a direct and logical method to define the equation of a line.

Transforming an equation into the slope-intercept form can provide a clearer picture of the line's behavior and it's easier to interpret graphically. The slope-intercept form of a linear equation is:\( y = mx + b \), where:

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, meaning it's the value where the line crosses the y-axis.

Starting with the point-slope form from the earlier task, it's a simple algebraic process to isolate \(y\) on one side of the equation. Once that's done, you are left with \(y\) equal to a term that includes \(x\), making it straightforward to see both the slope and y-intercept of your line.The slope-intercept form is not only a convenient form for graphing but also for quickly understanding how the line moves: the slope tells you the direction and steepness, while the intercept gives you an initial reference point on the y-axis.

Finding the slope of a line involves a straightforward calculation when you have two points. The formula for determining the slope \(m\) is:\( m = \frac{y_2 - y_1}{x_2 - x_1} \).This formula simply means you are comparing the change in \(y\)-values (vertical change) to the change in \(x\)-values (horizontal change) between two points on the line:

• \((x_1, y_1)\) is the first point.
• \((x_2, y_2)\) is the second point.

As demonstrated in the solution, if you know the \(y\)-intercept, like the point \((0, 2)\), and you have another point on the line, such as \((4, -1)\), you can substitute these into the slope formula. The calculation produces the line's rate of up or down movement.Once you find the slope, you can plug it back into the point-slope formula, allowing you to write the line's equation. Understanding how to find the slope is crucial, as it is the bridge between the concrete points on the coordinate plane and the abstract equation of the line.