Question
using x only
linear equations: one with infinite
solutions, one with no solution, and one with one solution. Write a real-world scenario
that goes with at least one system of equations. Show your work
linear equations: one with infinite
solutions, one with no solution, and one with one solution. Write a real-world scenario
that goes with at least one system of equations. Show your work
Answers
GPT-4o mini
Let’s create three distinct linear equations using the variable \( x \):
1. **Infinite Solutions**:
- Equation: \( 2x + 4 = 2(x + 2) \)
- Simplifying, we get:
\[
2x + 4 = 2x + 4 \quad \text{(true for all values of } x\text{)}
\]
2. **No Solution**:
- Equation: \( x + 3 = x - 1 \)
- Simplifying, we get:
\[
x + 3 - x = -1 \implies 3 = -1 \quad \text{(not possible)}
\]
3. **One Solution**:
- Equation: \( 3x - 5 = 4 \)
- Simplifying, we get:
\[
3x = 9 \implies x = 3 \quad \text{(unique solution)}
\]
### Real-World Scenario:
Let's consider a scenario involving a small business selling two types of products, A and B.
- **Product A** and **Product B** are sold for the same price of $2 each.
- **Scenario for Infinite Solutions**: If the business claims that the revenue from selling products A and B can be represented by the equation \( 2x + 4 = 2(x + 2) \), it indicates that no matter how many units they sell (as long as they sell multiples of 2), the revenue would always balance to the same equation—a statement that holds true for all quantities, hence infinite solutions.
- **Scenario for No Solution**: Let's consider a scenario where the business claims that they sell \( x + 3 \) units of product A and \( x - 1 \) units of product B. The equation \( x + 3 = x - 1 \) implies that the count of product A sold is equal to the count of product B sold with a discrepancy of 4 (3 vs -1). This leads to a contradiction, meaning it's impossible to satisfy both conditions at the same time, hence no solution.
- **Scenario for One Solution**: Suppose the business calculates that in order to achieve a revenue of $4 from selling products, they set up the condition that the sales of product A (priced at $2) and other incomings must satisfy \( 3x - 5 = 4 \). This indicates that they can solve for the necessary number of units \( x \) (which is 3 in this case) in order to yield their desired revenue target of $4. Since this equation leads to only one specific value for \( x \), it implies there’s only one feasible way to reach that goal.
### Summary of Equations with Context:
- Infinite Solutions: Suggest channels of sale that yield equal revenue across all variations of products sold.
- No Solution: Indicates an inconsistency in sales reporting that cannot logically coexist.
- One Solution: Reflects a specific outcome or target the business aims to achieve in terms of sales and revenue.
1. **Infinite Solutions**:
- Equation: \( 2x + 4 = 2(x + 2) \)
- Simplifying, we get:
\[
2x + 4 = 2x + 4 \quad \text{(true for all values of } x\text{)}
\]
2. **No Solution**:
- Equation: \( x + 3 = x - 1 \)
- Simplifying, we get:
\[
x + 3 - x = -1 \implies 3 = -1 \quad \text{(not possible)}
\]
3. **One Solution**:
- Equation: \( 3x - 5 = 4 \)
- Simplifying, we get:
\[
3x = 9 \implies x = 3 \quad \text{(unique solution)}
\]
### Real-World Scenario:
Let's consider a scenario involving a small business selling two types of products, A and B.
- **Product A** and **Product B** are sold for the same price of $2 each.
- **Scenario for Infinite Solutions**: If the business claims that the revenue from selling products A and B can be represented by the equation \( 2x + 4 = 2(x + 2) \), it indicates that no matter how many units they sell (as long as they sell multiples of 2), the revenue would always balance to the same equation—a statement that holds true for all quantities, hence infinite solutions.
- **Scenario for No Solution**: Let's consider a scenario where the business claims that they sell \( x + 3 \) units of product A and \( x - 1 \) units of product B. The equation \( x + 3 = x - 1 \) implies that the count of product A sold is equal to the count of product B sold with a discrepancy of 4 (3 vs -1). This leads to a contradiction, meaning it's impossible to satisfy both conditions at the same time, hence no solution.
- **Scenario for One Solution**: Suppose the business calculates that in order to achieve a revenue of $4 from selling products, they set up the condition that the sales of product A (priced at $2) and other incomings must satisfy \( 3x - 5 = 4 \). This indicates that they can solve for the necessary number of units \( x \) (which is 3 in this case) in order to yield their desired revenue target of $4. Since this equation leads to only one specific value for \( x \), it implies there’s only one feasible way to reach that goal.
### Summary of Equations with Context:
- Infinite Solutions: Suggest channels of sale that yield equal revenue across all variations of products sold.
- No Solution: Indicates an inconsistency in sales reporting that cannot logically coexist.
- One Solution: Reflects a specific outcome or target the business aims to achieve in terms of sales and revenue.