Question

This graph shows the inequality 25z + 15y ≥ 200, and represents Lina's earnings for lawn mowing and dog walking where x represents the number of lawns mowed and y represents the number of dogs walked. Which of the labeled points is a viable solution in this context?
(2,3)
None of the points are a viable solution
(20,11.5)
(10,10)

Answer 1

To determine which of the labeled points (if any) is a viable solution for the inequality \( 25z + 15y \geq 200 \), we first need to express the inequality in terms of the variables provided. In your context, it looks like there might be a mistake in the variable names—let's assume \( z \) should actually be \( x \) (the number of lawns mowed) and \( y \) (the number of dogs walked). Therefore, the inequality you want to check is:

\[ 25x + 15y \geq 200 \]

Next, we'll test each of the given points \( (2,3) \), \( (20, 11.5) \), and \( (10, 10) \):

1. **For (2, 3)**:
\[
25(2) + 15(3) = 50 + 45 = 95
\]
\[
95 \geq 200 \quad \text{(false)}
\]

2. **For (20, 11.5)**:
\[
25(20) + 15(11.5) = 500 + 172.5 = 672.5
\]
\[
672.5 \geq 200 \quad \text{(true)}
\]

3. **For (10, 10)**:
\[
25(10) + 15(10) = 250 + 150 = 400
\]
\[
400 \geq 200 \quad \text{(true)}
\]

Now, let's summarize the results:

- Point \( (2, 3) \): **Not a solution**
- Point \( (20, 11.5) \): **Viable solution**
- Point \( (10, 10) \): **Viable solution**

Given this analysis, both points \( (20, 11.5) \) and \( (10, 10) \) are viable solutions for the inequality \( 25x + 15y \geq 200 \).

Since the question asks for which of the labeled points is a viable solution in this context, it would be correct to say that **both (20, 11.5) and (10, 10) are viable solutions.** If you must choose just one point, then either valid choice can be considered correct depending on the context of the question.