Modeling Word Problems with Elimination

Property To solve a word problem using the elimination method: 1. Define two variables for the unknown quantities. 2. Write a system of two linear equations that models the conditions. 3. Solve the system using elimination. 4. Interpret the solution $(x, y)$ in the context of the original problem.

Examples Bakery: 2 muffins ($m$) and 3 bagels ($b$) cost $12, while 2 muffins and 1 bagel cost $8. System: $2m + 3b = 12$ and $2m + b = 8$. Subtract the second equation from the first to eliminate $m$: $2b = 4$, so $b = 2$. Substitute $b = 2$ into the second equation: $2m + 2 = 8$, which means $m = 3$. Muffins cost $3 and bagels cost $2. Theater Tickets: A theater sold 100 tickets total. Adult tickets ($a$) cost $5 and child tickets ($c$) cost $3, with total sales of $340. System: $a + c = 100$ and $5a + 3c = 340$. Multiply the first equation by 3 to get $ 3a 3c = 300$. Add this to the second equation to eliminate $c$: $2a = 40$, so $a = 20$. Substitute $a = 20$ into the first equation to find $c = 80$. They sold 20 adult and 80 child tickets.

Explanation Many real world situations involve two unknown quantities that relate to each other in two different ways. By defining a variable for each unknown, you can write a system of two linear equations. Elimination is incredibly efficient for word problems because the equations often naturally form in Standard Form ($Ax + By = C$) based on how quantities and prices are described in the text.