How can we tell that the line y = x passes through the origin, has a slope of 1, and makes a 45-degree angle with each axis? I’m struggling to understand symmetry with respect to the line with equation y = x.

Let us consider different values of x, and find y.

- For x=0, y=0
- For x=1, y=1
- For x=2, y=2
- For x=-1, y=-1
- For x=3.14, y=3.14

When we draw a straight line through all these points, we can see that the line passes through (x,y)=(0,0). In other words, it passes through the origin.

When a line is denoted in the form of y=mx+c, c and m are constants where m denotes the slope. For line y=x, we can see that m=1, and c = 0, so the slope is 1.

Thank you so much. I think I am missing the foundation to even interpret it.

Do you always check whether the line passes through (x,y) = (0,0) in all these equations?

Just substitute a few values for x and find the corresponding y values. If one of the values you choose for x is x=0 and find y. That y value is known as the y intercept (or where the line crosses the y axis)… and vice versa. For the very basics of visualising a line, just go through this video

If it is explicitly asked in the question “does the line pass through the origin”, you can verify it by substituting x=0 and if you find that for x=0, y also turns out to be 0, you can say that the line passes through the origin

Thank you so much. This is really helpful.