Question

a homeowner is worried about the work needed to maintain a grass lawn and flower beds, so she is now looking at some low - maintenance materials. she is considering artificial turf, which costs $15 per square foot to install, and gravel, which costs $3 per square foot. she may use a combination of the two materials in different parts of the yard. her budget is $3,000. here is a graph representing some constraints in the situation.
part 1: to answer questions 1 - 2, use the equation $3x + 15y = 3000$, where x represents the gravel and y represents the artificial turf.

1. in this situation, what does the point $(500,100)$ mean?
2. does $(500,100)$ represent a solution to the equation? explain or show your reasoning.

Explanation:

Step1: Understand the meaning of variables

In the equation \(3x + 15y=3000\), \(x\) represents the area (in square - feet) of gravel and \(y\) represents the area (in square - feet) of artificial turf.

Step2: Analyze the point \((500,100)\) for meaning

The \(x\) - coordinate \(x = 500\) means the area of gravel is \(500\) square feet. The \(y\) - coordinate \(y = 100\) means the area of artificial turf is \(100\) square feet.

Step3: Check if \((500,100)\) is a solution

Substitute \(x = 500\) and \(y = 100\) into the left - hand side of the equation \(3x+15y\).

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Answer:

1. The point \((500,100)\) means that the homeowner uses \(500\) square feet of gravel and \(100\) square feet of artificial turf.
2. Yes, \((500,100)\) is a solution to the equation \(3x + 15y=3000\) because when \(x = 500\) and \(y = 100\), \(3x+15y=3\times500 + 15\times100=1500+1500 = 3000\), which is equal to the right - hand side of the equation.