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Understanding the slope-intercept form of a linear equation is essential in graphing. The slope-intercept form is written as \(y = mx + b\). Here, \(m\) represents the slope of the line — how steep it is — and \(b\) indicates where the line crosses the y-axis (the y-intercept).

By rewriting the equation \(-x + y = 3\) into slope-intercept form, we isolate \(y\):

\[ y = x + 3 \]
In this case, \(m = 1\) and \(b = 3\). Knowing this helps us quickly understand the behavior of the line: it rises 1 unit for every 1 unit it moves to the right, and it crosses the y-axis at \(y = 3\).

Plotting points is a practical step toward graphing a linear equation. To plot points, you need pairs of \(x\) and \(y\) values that satisfy the equation. For the equation \(y = x + 3\), choose a few \(x\) values and find the corresponding \(y\) values.

Here is an example:

• For \(x = -2\), \(y = -2 + 3 = 1\)
• For \(x = -1\), \(y = -1 + 3 = 2\)
• For \(x = 0\), \(y = 0 + 3 = 3\)
• For \(x = 1\), \(y = 1 + 3 = 4\)
• For \(x = 2\), \(y = 2 + 3 = 5\)

Once you have these (x, y) pairs, you can plot them on the Cartesian plane. This reveals the line's shape.

The Cartesian plane is a two-dimensional surface defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). It is used for plotting points, lines, and curves.

Each point on the plane is represented by a pair of coordinates \((x, y)\). The first number is the x-coordinate, indicating the horizontal position, and the second number is the y-coordinate, indicating the vertical position.

For example, the point \((-2, 1)\) lies 2 units to the left of the y-axis and 1 unit above the x-axis. Similarly, the point \((0, 3)\) is on the y-axis itself, 3 units above the x-axis. By plotting all our calculated points and connecting them with a straight line, we graph the equation on this plane.