given that the graph of y=kx passes through (a,b) and a/b=2, then what does k equal?

Given that the graph of \( y = kx \) passes through the point \((a, b)\) and \( \frac{a}{b} = 2 \), we need to determine the value of \( k \).

1. Since the point \((a, b)\) lies on the line \( y = kx \), we can substitute \( x = a \) and \( y = b \) into the equation:
   \[
   b = ka
   \]

2. We are also given that \( \frac{a}{b} = 2 \). This implies:
   \[
   a = 2b
   \]

3. Substitute \( a = 2b \) into the equation \( b = ka \):
   \[
   b = k(2b)
   \]

4. Simplify this equation:
   \[
   b = 2kb
   \]

5. To solve for \( k \), divide both sides by \( b \) (assuming \( b \neq 0 \)):
   \[
   1 = 2k
   \]

6. Finally, solve for \( k \):
   \[
   k = \frac{1}{2}
   \]

### Answer:
The value of \( k \) is \( \frac{1}{2} \).

Would you like more details, or do you have any questions?

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Here are some related questions:
1. What happens to the line's slope if \( \frac{a}{b} = \frac{1}{2} \) instead?
2. How does the line \( y = kx \) change if \( k = -\frac{1}{2} \)?
3. What is the significance of \( k \) when \( y = kx \) passes through the origin?
4. How would the equation change if the point given was in the form \( (-a, -b) \)?
5. What happens to \( k \) if the ratio \( \frac{a}{b} \) changes?

**Tip:** When given points or ratios, substituting them directly into the equation can simplify solving for unknown variables.

Learn how to determine the value of k for the equation y = kx, given that the line passes through a point (a, b) where the ratio a/b = 2. This problem involves substituting values, solving linear equations, and understanding ratios in mathematics. Suitable for students in Grades 7-9.

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