Part a:

The graphs of the equations $y = x$, $y = 3x$, and $y = -2x$ all pass through the origin $(0, 0)$ because:

1. The general form of a linear equation is $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.
2. In the equations $y = x$, $y = 3x$, and $y = -2x$, the y-intercept $c = 0$.
3. The y-intercept is the point where the graph crosses the y-axis, which happens when $x = 0$. For these equations, when $x = 0$, $y = 0$ as well, so the point $(0, 0)$ is on all the graphs.

Thus, all these lines pass through the origin because their y-intercepts are zero.

Part b:

The graphs of the equations $y = x - 1$, $y = 3x - 2$, and $y = 5 - 2x$ do not pass through the origin $(0, 0)$ because:

1. In each of these equations, there is a non-zero y-intercept $c$. Specifically:
   • For $y = x - 1$, the y-intercept is $c = -1$.
   • For $y = 3x - 2$, the y-intercept is $c = -2$.
   • For $y = 5 - 2x$, the y-intercept is $c = 5$.
2. When $x = 0$ in each of these equations, the corresponding $y$-value is not 0:
   • For $y = x - 1$, when $x = 0$, $y = -1$.
   • For $y = 3x - 2$, when $x = 0$, $y = -2$.
   • For $y = 5 - 2x$, when $x = 0$, $y = 5$.

Therefore, none of these graphs pass through the origin because their y-intercepts are not zero, meaning they cross the y-axis at points other than $(0, 0)$.

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Do you want further details or have any other questions?

Here are 5 related questions to expand on this:

1. What is the significance of the slope $m$ in the equation $y = mx + c$?
2. How would the graph change if the slope in $y = mx + c$ were zero?
3. Can you explain what it means when the y-intercept $c$ is positive or negative?
4. How do you determine the x-intercept of a linear equation?
5. How would you graph a line with a negative slope?

Tip: A linear equation’s y-intercept tells you where the line crosses the y-axis. If $c = 0$, the line will always pass through the origin.