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The point-slope formula is an essential tool for graphing linear equations. It's a straightforward method to find the equation of a line when you know a point on the line and the slope. The formula is:
\[ y - y_1 = m(x - x_1) \]
Here:

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

The point-slope form can be quickly transformed into other forms of a linear equation by simplifying it. By distributing and combining like terms, as shown in the example, the point-slope equation can easily become the slope-intercept form \(y = mx + b\), making it straightforward to graph. This approach ties together the geometric understanding of a line with its algebraic representation, allowing you to see how the slope and a specific point align on the coordinate plane.

Understanding the slope of a line is crucial in graphing and analyzing linear equations. The slope, often denoted by \(m\), is a measure of how steep the line is. It's calculated as the "rise" over the "run," or more technically, the change in \(y\) over the change in \(x\):
\[ m = \frac{\Delta y}{\Delta x} \]
Here's what that means in practical terms:

• If the slope is positive, as in our example where \(m = 3\), the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• A zero slope means the line is horizontal, while an undefined slope indicates a vertical line.

Slope connects the algebraic and graphical interpretations of a line, acting as a bridge between equations and visual representations. In graphing, it allows you to start at a given point and predict where the line runs next, as we did by moving up 3 units and 1 unit to the right.

The coordinate plane is a two-dimensional surface where we can graph equations, like the linear equation in our example. It consists of two perpendicular axes:

• The horizontal axis, usually called the \(x\)-axis
• The vertical axis, known as the \(y\)-axis

Every point on this plane can be represented by a pair of numbers \((x, y)\), which express how far along each axis the point is from the origin \((0,0)\).
In the context of graphing a line, the coordinate plane allows us to visually connect different mathematical elements:

• The point \((-5,1)\) is plotted as a specific location where the line will pass through.
• Using the slope, additional points like \((-4,4)\) can be found and plotted, ensuring the line is linear and accurate.

The coordinate plane is not just a visual aid but a powerful tool that connects algebraic equations to their geometric representations, making math both understandable and tangible.