To solve the problem, we need to create a linear equation in standard form that represents the gardener's budget for vinca and phlox plants.

Part (a): Formulating the Equation

Let's denote:

• $x$: the number of vinca plants
• $y$: the number of phlox plants

From the information provided:

• The maximum number of vinca plants the gardener can afford is 125.
• The maximum number of phlox plants is 60.

If we assume the total cost of all plants can be represented by a fixed budget (let's say 300), we can write the equation as follows:

$\frac{x}{125} + \frac{y}{60} = 1$

To convert this into standard form, we can multiply through by the least common multiple (LCM) of 125 and 60, which is 750. This leads to:

$6x + 12.5y = 750$

To avoid decimals, we can multiply the entire equation by 2:

$12x + 25y = 1500$

This is the equation in standard form that models the combinations of vinca and phlox plants.

Part (b): Graphing the Equation

To graph this equation:

1. Find the intercepts:

   • x-intercept (set $y = 0$): $12x = 1500 \implies x = 125$
   • y-intercept (set $x = 0$): $25y = 1500 \implies y = 60$

2. Plot the points:

   • (125, 0) for the x-intercept
   • (0, 60) for the y-intercept

3. Draw the line connecting these two points.

Part (c): Possible Combinations

Using the equation $12x + 25y = 1500$, we can find combinations of $x$ and $y$:

1. (0, 60): 0 vinca and 60 phlox
2. (125, 0): 125 vinca and 0 phlox
3. (72, 100): This combination is incorrect as it exceeds the limits; the correct combinations are:
   • (96, 50): 96 vinca and 50 phlox
   • (50, 96): 50 vinca and 96 phlox
4. (150, 48): Not valid since it exceeds the maximum vinca plants.

Valid Combinations

Here are four valid combinations:

1. 0 vinca and 60 phlox
2. 125 vinca and 0 phlox
3. 96 vinca and 50 phlox
4. 50 vinca and 96 phlox

Would you like further clarification on any part of this? Here are some additional questions to consider:

1. How would the equation change if the maximum number of plants changed?
2. Can you explain how to find other combinations along the line?
3. What would the slope of the line represent in this context?
4. How would you adjust the graph if the budget increased?
5. What real-life factors could affect the number of plants a gardener can buy?