Question
What are some real-world examples that model the equation −x+40=x?
Answers
GPT-4o mini
The equation \(-x + 40 = x\) is a simple linear equation that can be solved to find the value of \(x\). Solving it, we combine like terms:
\[
-x + 40 = x
\]
Add \(x\) to both sides:
\[
40 = 2x
\]
Now divide by 2:
\[
x = 20
\]
This means that \(x = 20\) is the solution to the equation.
Now, let's look at some real-world examples that can be modeled by this equation:
1. **Budgeting**: Suppose you have $40 and plan to spend an equal amount \(x\) on two different activities. The situation can be modeled by the equation: the amount you're going to spend on one activity, which is \(x\), plus the amount you're going to spend on the other activity, which is represented as \(-x + 40\) (the remaining money after spending \(x\)), should equal your planned spending of \(x\). If you solve for \(x\), you find that you should be spending $20 on each activity.
2. **Distance and Speed**: Imagine two cars start from the same point and drive towards each other. Car A drives towards Car B at a speed of \(x\) miles per hour, while Car B drives away from Car A at a speed that means after a certain time they will be 40 miles apart. If the car driving away travels at a speed of \(-x + 40\) miles per hour, the equation describes how the distances covered by both cars relate over time until they are 40 miles apart. Solving the equation gives the speed for both cars.
3. **Workload Distribution**: Consider a scenario at work where two employees are tasked with completing a project together. Employee A can complete a portion of the project in \(-x + 40\) hours, while Employee B works at a rate of \(x\) hours. The equation models that the total workload they can handle is equal, balancing their efforts at 20 hours each to complete the project together.
4. **Cooking**: Suppose you have 40 ingredients and need to assign \(x\) ingredients to two dishes. If one dish uses all the remaining ingredients after taking \(x\), then the scenario can be described by the equation. In this case, \(x = 20\) implies each dish would use 20 ingredients.
These examples illustrate situations where the linear equation \(-x + 40 = x\) models real-world scenarios, making the solution \(x = 20\) practically meaningful.
\[
-x + 40 = x
\]
Add \(x\) to both sides:
\[
40 = 2x
\]
Now divide by 2:
\[
x = 20
\]
This means that \(x = 20\) is the solution to the equation.
Now, let's look at some real-world examples that can be modeled by this equation:
1. **Budgeting**: Suppose you have $40 and plan to spend an equal amount \(x\) on two different activities. The situation can be modeled by the equation: the amount you're going to spend on one activity, which is \(x\), plus the amount you're going to spend on the other activity, which is represented as \(-x + 40\) (the remaining money after spending \(x\)), should equal your planned spending of \(x\). If you solve for \(x\), you find that you should be spending $20 on each activity.
2. **Distance and Speed**: Imagine two cars start from the same point and drive towards each other. Car A drives towards Car B at a speed of \(x\) miles per hour, while Car B drives away from Car A at a speed that means after a certain time they will be 40 miles apart. If the car driving away travels at a speed of \(-x + 40\) miles per hour, the equation describes how the distances covered by both cars relate over time until they are 40 miles apart. Solving the equation gives the speed for both cars.
3. **Workload Distribution**: Consider a scenario at work where two employees are tasked with completing a project together. Employee A can complete a portion of the project in \(-x + 40\) hours, while Employee B works at a rate of \(x\) hours. The equation models that the total workload they can handle is equal, balancing their efforts at 20 hours each to complete the project together.
4. **Cooking**: Suppose you have 40 ingredients and need to assign \(x\) ingredients to two dishes. If one dish uses all the remaining ingredients after taking \(x\), then the scenario can be described by the equation. In this case, \(x = 20\) implies each dish would use 20 ingredients.
These examples illustrate situations where the linear equation \(-x + 40 = x\) models real-world scenarios, making the solution \(x = 20\) practically meaningful.