To represent the situation mathematically, we can express the costs of installing grass sod and flower beds with the following inequality:
\[ 5x + 10y \leq 4500 \]
where:
- \( x \) is the area in square feet of grass sod to be installed,
- \( y \) is the area in square feet of flower beds to be installed.
1. Graph the Inequality
To graph the inequality, we first rewrite it as an equation to find the boundary line:
\[ 5x + 10y = 4500 \]
Next, we can rearrange it to get \( y \) in terms of \( x \):
\[ 10y = 4500 - 5x \] \[ y = 450 - 0.5x \]
Now we can find the x- and y-intercepts to graph the equation.
-
Y-intercept: Set \( x = 0 \): \[ y = 450 - 0.5(0) = 450 \quad \text{(Point: } (0, 450) \text{)} \]
-
X-intercept: Set \( y = 0 \): \[ 0 = 450 - 0.5x \implies 0.5x = 450 \implies x = 900 \quad \text{(Point: } (900, 0) \text{)} \]
Now we plot these intercepts on a coordinate graph and draw the line connecting them. The area below this line represents the solutions to the inequality \( 5x + 10y \leq 4500 \).
2. Choose Three Points that are Solutions
Below are three points that satisfy the inequality, along with their meanings in the context of the situation:
-
Point (0, 400):
- Here, \( x = 0 \) (no grass sod installation), and \( y = 400 \) (400 square feet of flower beds). The total cost is \( 5(0) + 10(400) = 4000 \), which is within the budget. This means Michelle can choose to fill the entire budget with only flower beds.
-
Point (900, 0):
- Here, \( x = 900 \) (900 square feet of grass sod) and \( y = 0 \) (no flower beds). The total cost is \( 5(900) + 10(0) = 4500 \), which exactly meets the budget. This indicates that Michelle can opt for the maximum area of grass sod within her budget without spending on flower beds.
-
Point (450, 225):
- Here, \( x = 450 \) (450 square feet of grass sod) and \( y = 225 \) (225 square feet of flower beds). The total cost is \( 5(450) + 10(225) = 2250 + 2250 = 4500 \), which also meets the budget. This option shows a balanced approach where Michelle can achieve both grass and flower beds while staying within budget.
These points exemplify the different combinations of grass sod and flower bed installations that Michelle can consider for her landscaping project while adhering to her budget constraints.