Slope and intercept
Slide m and b and watch the line y = mx + b move. Then grab either of the two dots and drag it along the line. The triangle between the dots shows the rise and the run, and the readout divides one by the other. Notice that wherever you put the dots, rise ÷ run comes out as m. Switch on the parallel line and compare its equation with the first one.
What's going on
A line is the graph of a steady rate. Every time x goes up by 1, y goes up by the same amount, and that amount is the slope m. Between any two points on the line the rise is m times the run, so rise ÷ run is always m, no matter which two points you pick. That is why one number describes the steepness of the whole line. The triangles you make by dragging the dots are all similar: same shape, different size.
The intercept b is where the line crosses the y-axis, because putting x = 0 into y = mx + b leaves y = b. Changing b slides the line up or down without tilting it. Changing m tilts the line about the point (0, b), which stays fixed.
Two lines with the same m and different b are parallel: the gap between them is the same everywhere, so they never meet. If m = 0 the line is flat and y = b for every x. There is no slider for a vertical line because its slope would be rise ÷ 0, which is not a number. A vertical line is x = c, not y = mx + b.
Sheets: Why slope is the same everywhere on a line and Reading a line at sight.