Math Problem Statement

Suppose you can spend no more than 15 hours a week at your two jobs. Mowing lawns pays $3 an hour and babysitting pays $5 an hour. You need to earn at least $60 a week. Write and graph a system of linear inequalities and write two possible solutions.

Solution

Let's analyze the problem and construct the system of inequalities, followed by solving for possible solutions.

Problem Summary

You have two jobs:

  • Mowing lawns which pays $3 per hour.
  • Babysitting which pays $5 per hour.

You have two conditions:

  1. You can work no more than 15 hours a week between both jobs.
  2. You need to earn at least $60 a week.

Let:

  • xx = hours spent mowing lawns
  • yy = hours spent babysitting

Step 1: Write the System of Inequalities

  1. Time constraint: The total hours spent on both jobs should be no more than 15 hours. x+y≤15x + y \leq 15

  2. Earnings constraint: The total earnings from both jobs should be at least $60.

    • Earnings from mowing lawns: 3x3x
    • Earnings from babysitting: 5y5y 3x+5y≥603x + 5y \geq 60

So, the system of inequalities is:

x + y \leq 15 \\ 3x + 5y \geq 60 \end{cases}$$ ### Step 2: Write Two Possible Solutions To find two possible solutions, we need values of $$x$$ and $$y$$ that satisfy both inequalities. #### Possible Solution i Let’s try $$x = 5$$ and $$y = 10$$: 1. Check the time constraint: $$5 + 10 = 15$$ (satisfied). 2. Check the earnings constraint: $$3(5) + 5(10) = 15 + 50 = 65 \geq 60$$ (satisfied). Thus, $$(x, y) = (5, 10)$$ is a valid solution. #### Possible Solution ii Let’s try $$x = 10$$ and $$y = 5$$: 1. Check the time constraint: $$10 + 5 = 15$$ (satisfied). 2. Check the earnings constraint: $$3(10) + 5(5) = 30 + 25 = 55 \geq 60$$ (not satisfied). This point does not work, so let's try another. #### Possible Solution iii Let’s try $$x = 8$$ and $$y = 7$$: 1. Check the time constraint: $$8 + 7 = 15$$ (satisfied). 2. Check the earnings constraint: $$3(8) + 5(7) = 24 + 35 = 59$$ (not satisfied). We’ll need to try another combination. A feasible one is: #### Possible Solution iv Try $$x = 10$$ and $$y = 10$$: 1.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Inequalities
Systems of Inequalities

Formulas

Time constraint: x + y <= 15
Earnings constraint: 3x + 5y >= 60

Theorems

-

Suitable Grade Level

Grades 8-10

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