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When finding the equation of a line, the first step is to calculate the slope. The slope, represented by the letter "m," measures how steep the line is. Imagine climbing a hill; the steeper the hill, the larger the slope. In mathematical terms, slope describes the rate of change between two points on a line. To find the slope, use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]Here, \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of two specific points on the line. In our example, the points are \((-4, 7)\) and \(2, 4)\). By substituting the values, we have:\[m = \frac{4 - 7}{2 - (-4)} = \frac{-3}{6} = -0.5\]This means for every unit you move to the right along the x-axis, you move half a unit down along the y-axis. Understanding this concept is crucial, as it sets the stage for formulating the line's equation.

The y-intercept of a line is a fundamental concept that describes where the line crosses the y-axis. This point is important because it gives us specific information about the line's position. The y-intercept is represented by the letter "b" in the line's equation \(y = mx + b\).To find the y-intercept, we continue from where we calculated the slope. Using the slope \(m = -0.5\) and one of the given points \((2, 4)\), substitute these values into the equation:\[4 = -0.5 \times 2 + b\]By solving this equation, we isolate "b" to find:\[b = 4 + 1 = 5\]This tells us that the line crosses the y-axis at the point \((0, 5)\). Recognizing the y-intercept is key to drawing the line accurately and fully understanding its equation.

Now that we have both the slope and the y-intercept, we can form a complete equation of the line. The equation follows the format:\[y = mx + b\]Where:

• "m" is the slope.
• "b" is the y-intercept.

For our specific case, we've found \(m = -0.5\) and \(b = 5\). Insert these values into the equation format to get:\[y = -0.5x + 5\]This is the final equation of the line that passes through the points \((-4, 7)\) and \(2, 4)\). Understanding how each part of the equation fits together—how the slope impacts the line's tilt and the y-intercept determines its vertical position—is crucial to mastering linear equations.