Integrated Math 1 / Unit 6: Systems of Equations and Inequalities
6.5 Modeling with Systems
Notes
To model a situation, define two variables, write two equations, and solve.
- Typical setups: total count plus total value; two plans with different fees; sum and difference of numbers.
- Example: $x + y = 20$ and $x - y = 6$ give $x = 13$ and $y = 7$.
- Interpret the answer in the story and check units.
- Make sure your answer makes sense (no negative people).
Practice
10 tickets were sold for a total of $65. Adult tickets cost $8 and child tickets cost $5. How many adult tickets were sold?
$a + c = 10$ and $8a + 5c = 65$. Substituting $c = 10 - a$: $3a + 50 = 65$, so $a = 5$.
Plumber A charges $40 plus $30 per hour. Plumber B charges $70 plus $20 per hour. After how many hours do they cost the same?
$40 + 30h = 70 + 20h \Rightarrow 10h = 30 \Rightarrow h = 3$. Both cost $130.
Two numbers have a sum of 20 and a difference of 6. What is the larger number?
$x + y = 20$ and $x - y = 6$ give $2x = 26$, so $x = 13$.
Ready for more? Take the Unit 6 test.