Systems of Equations and Word Problem Modeling

Systems of Equations and Word Problem Modeling

A system of equations consists of two or more equations with shared variables. The solution is the ordered pair $(x, y)$ that satisfies both equations simultaneously (the point of intersection on a graph).

Solution Methods

1. Substitution: Isolate one variable in one equation and substitute its expression into the other.
2. Elimination (Addition/Subtraction): Multiply one or both equations by constants so that adding the equations eliminates one variable.

Worked Example:

A movie theater sold 200 tickets for a total of $1,500. Adult tickets cost $10 and student tickets cost $5. How many adult tickets were sold?

1. Define variables:
   • $a$ = number of adult tickets
   • $s$ = number of student tickets
2. Set up equations: $$a + s = 200$$ $$10a + 5s = 1500$$
3. Multiply the first equation by -5: $$-5a - 5s = -1000$$ $$10a + 5s = 1500$$
4. Add the equations: $$5a = 500 \implies a = 100$$
5. There were 100 adult tickets sold.