Given the linear equations y = x, y = 2x, and y = 2x + 1, determine the gradient and y-intercept for each, plot the graphs, and identify the gradients of the three graphs.

The standard equation of a linear graph is given by:

where $m$ is the gradient and $c$ is the y-intercept.

Complete the table for the gradient and y-intercept for each equation:

The gradients of the 3 linear graphs are $\boxed{\text{1, 2, 2}}$.

To plot the 3 graphs, find at least two points for each equation and draw a straight line through them.

• For $y = x$:

  • If $x = 0$, $y = 0$. Point: $(0, 0)$
  • If $x = 2$, $y = 2$. Point: $(2, 2)$
  • Plot these points and draw a line through them.

• For $y = 2x$:

  • If $x = 0$, $y = 0$. Point: $(0, 0)$
  • If $x = 2$, $y = 4$. Point: $(2, 4)$
  • Plot these points and draw a line through them.

• For $y = 2x + 1$:

  • If $x = 0$, $y = 1$. Point: $(0, 1)$
  • If $x = 2$, $y = 5$. Point: $(2, 5)$
  • Plot these points and draw a line through them.