Linear equations  (Page 2/6)

The graph of the line $x=-2$ is a vertical line that has the x-coordinate –2 no matter what the y-coordinate is. Therefore, the graph is a vertical line passing through (–2, 0).

The graph of the line $y=3$ , is a horizontal line that has the y-coordinate 3 regardless of what the x-coordinate is. Therefore, the graph is a horizontal line that passes through (0, 3).

Let $(x_1,y_1)=(-\mathrm{2,3})$ and $(x_2,y_2)=(\mathrm{4,}-1)$ then the slope

To give the reader a better understanding, both the vertical change, –4, and the horizontal change, 6, are shown in the above figure.

When two points are given, it does not matter which point is denoted as $(x_1,y_1)$ and which $(x_2,y_2)$ . The value for the slope will be the same. For example, if we choose $(x_1,y_2)=(\mathrm{4,}-1)$ and $(x_2,y_2)=(-\mathrm{2,3})$ , we will get the same value for the slope as we obtained earlier. The steps involved are as follows.

The student should further observe that if a line rises when going from left to right, then it has a positive slope; and if it falls going from left to right, it has a negative slope.

Let $(x_1,y_1)=(\mathrm{2,3})$ and $(x_2,y_2)=(\mathrm{2,}-1)$ then the slope

Slope equals $\frac{\text{rise}}{\text{run}}$ . The fact that the slope is $\frac{-3}{4}$ , means that for every rise of –3 units (fall of 3 units) there is a run of 4. So if from the given point (1, 2) we go down 3 units and go right 4 units, we reach the point (5, –1). The following graph is obtained by connecting these two points.

Alternatively, since $\frac{3}{-4}$ represents the same number, the line can be drawn by starting at the point (1,2) and choosing a rise of 3 units followed by a run of –4 units. So from the point (1, 2), we go up 3 units, and to the left 4, thus reaching the point (–3, 5) which is also on the same line. See figure below.