Consider the following Budget Lines:

BL1 : 3x + 13y = 27

BL2: 6x + 26y = 43 , The slope of BL 1 is (calculate slope as dy/dx)


a.
3/13


b.
1/9


c.
13/3


d.
9


e.
1/6

The budget line equation is generally given in the form of \( Ax + By = C \). To find the slope of the budget line, we rearrange this equation in the form \( y = mx + b \), where \( m \) is the slope.

### Step 1: Rearrange BL1 into slope-intercept form
Starting with the equation of BL1:
\[
3x + 13y = 27
\]

Subtract \( 3x \) from both sides:
\[
13y = -3x + 27
\]

Now, divide the entire equation by 13 to solve for \( y \):
\[
y = -\frac{3}{13}x + \frac{27}{13}
\]

### Step 2: Identify the slope
The slope \( m \) is the coefficient of \( x \). Here, the slope \( m \) is \( -\frac{3}{13} \).

Since we are asked for the slope as \( \frac{dy}{dx} \), and the options only contain positive values, we take the magnitude of the slope:
\[
\frac{dy}{dx} = \frac{3}{13}
\]

### Conclusion:
The correct answer is:

**a. \( \frac{3}{13} \)**

Would you like further details or have any questions? Here are some related questions to expand your understanding:

1. What would be the slope if the equation of BL1 was \( 13x + 3y = 27 \)?
2. How does the slope of BL2 compare to the slope of BL1?
3. What is the intercept of BL1 on the x-axis?
4. If the prices of goods represented by \( x \) and \( y \) change, how does that affect the slope of the budget line?
5. What would the equation of BL1 look like if it was shifted upwards by 10 units?

**Tip:** Remember that the slope of a line gives you the rate of trade-off between the two goods in a budget constraint.

Learn how to calculate the slope of budget line BL1: 3x + 13y = 27 step-by-step. Understand the concepts of linear equations, slopes, and their applications. Suitable for high school students.

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