Real-world examples of systems of equations with infinite solutions

📚 Quick Study Guide

• 🔢 A system of equations has infinite solutions when the equations represent the same line.
• 📝 This means one equation is a multiple of the other, or they can be manipulated to look identical.
• 📈 Graphically, both equations would plot as the same line.
• 🧮 Algebraically, solving one equation for a variable and substituting into the other results in an identity (e.g., $0 = 0$).
• 💡 For a system of two linear equations, if the ratio of the coefficients of $x$ and $y$ and the constant terms are equal, then there are infinite solutions. For example, $ax + by = c$ and $kax + kby = kc$.

🧪 Practice Quiz

1. Which of the following real-world scenarios could be modeled by a system of equations with infinite solutions?
   1. A) Determining the cost of apples and oranges given two different purchase amounts.
   2. B) Calculating the speed of a boat in still water and the speed of the current.
   3. C) Finding two numbers whose sum is 10 and whose difference is 2.
   4. D) Determining the amount of time it takes for two people to paint a room together when they paint at the same rate.
2. Which system of equations has infinite solutions?
   1. A) $x + y = 5$, $x - y = 1$
   2. B) $2x + 2y = 4$, $x + y = 2$
   3. C) $x + y = 3$, $x + y = 5$
   4. D) $x - y = 1$, $x + y = 1$
3. In a business, the cost of producing $x$ items is given by $C = 2x + 5$. The revenue from selling $x$ items is given by $R = 2x + 5$. What does this imply?
   1. A) The business will always make a profit.
   2. B) The business will always lose money.
   3. C) The business will break even for any number of items produced and sold.
   4. D) The business needs more information to determine profitability.
4. A recipe calls for twice as much sugar as flour. Which system models this if $s$ is sugar and $f$ is flour and there is no other constraint?
   1. A) $s = 2f$
   2. B) $f = 2s$
   3. C) $s + f = 3$
   4. D) $s - f = 1$
5. Two different measurements are taken of the length of a garden. One measurement says it's $L$ and the other says it's $L$. This can be modeled as:
   1. A) $L = L$
   2. B) $L + L = 2$
   3. C) $L - L = 1$
   4. D) $L = 0$
6. Which of the following parameterizations represents a system with infinite solutions for the lines $x + y = a$ and $2x + 2y = b$?
   1. A) $a = 1, b = 1$
   2. B) $a = 2, b = 2$
   3. C) $a = 1, b = 2$
   4. D) $a = 2, b = 1$
7. Consider the equations $3x + 6y = 9$ and $x + 2y = 3$. What can be concluded about the solution set?
   1. A) There is a unique solution.
   2. B) There are infinite solutions.
   3. C) There is no solution.
   4. D) The solution is undefined.