Where does Jeremiah get the equation ' \( 21 n=45+18 n \) ' from?

Jeremiah likely derived the equation \( 21n = 45 + 18n \) from a problem that involves setting two expressions equal to each other based on specific conditions or relationships described in that problem. While I don't have the exact context of Jeremiah's situation, I can provide a general explanation of how such an equation might arise. ### Common Scenarios Leading to \( 21n = 45 + 18n \) 1. **Comparing Two Quantities:** - **Example:** Suppose Jeremiah is comparing the total cost of two different plans. - **Plan A:** Costs \$21 per item, so the total cost is \( 21n \), where \( n \) is the number of items. - **Plan B:** Has a fixed fee of \$45 plus \$18 per item, so the total cost is \( 45 + 18n \). - **Setting Them Equal:** To find out when both plans cost the same, Jeremiah sets the total costs equal: \[ 21n = 45 + 18n \] 2. **Balancing Equations in Mixture Problems:** - **Example:** If Jeremiah is mixing two solutions with different concentrations and wants the final mixture to have a specific concentration, he might set up an equation where the total amounts of a particular substance from both solutions equal the desired amount in the mixture. - The equation could look something like: \[ 21n = 45 + 18n \] where each term represents the amount of the substance from each part of the mixture. 3. **Solving for Equilibrium in Financial or Physical Problems:** - **Example:** In a job where Jeremiah earns a commission, he might set his earnings from this commission equal to a base salary plus a different commission structure. - The equation: \[ 21n = 45 + 18n \] could represent equating two different earning scenarios to find the number of units \( n \) needed to reach equilibrium. ### Solving the Equation To provide further clarity, here's how Jeremiah would solve the equation: \[ 21n = 45 + 18n \] 1. **Subtract \( 18n \) from both sides to gather like terms:** \[ 21n - 18n = 45 \] \[ 3n = 45 \] 2. **Divide both sides by 3 to solve for \( n \):** \[ n = \frac{45}{3} = 15 \] So, \( n = 15 \). This means that under the given conditions, whatever \( n \) represents (e.g., number of items, hours worked, etc.), the value is 15. ### Conclusion Without specific details about Jeremiah's problem, I can only speculate on the exact origin of the equation. However, equations of this form are commonly used in various algebraic contexts where two expressions involving the same variable are set equal to find the value of that variable. If you provide more details about the original problem Jeremiah is solving, I can offer a more precise explanation.