10 a
The graphs of y = x, y = 3x and y = -2x all pass through the origin (0, 0). Explain why. b The graphs of y = x - 1, y = 3x - 2 and y = 5 - 2x do not pass through the origin (0, 0). Explain why.

### 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.

The graphs of y = x, y = 3x and y = -2x all pass through the origin (0, 0). Explain why. The graphs of y = x - 1, y = 3x - 2 and y = 5 - 2x do not pass through the origin (0, 0). Explain why.

Explore why certain linear equations like y = x and y = 3x pass through the origin while others like y = x - 1 do not. Learn key algebraic concepts including the slope-intercept form, the role of the y-intercept, and how to graph linear equations. Ideal for students in Grades 8-10.

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