Systems of equations
Drag a point until it sits on two lines at once and discover that the crossing solves both equations.
Standards: 8.EE.C.8
1Wonder
Ana has 10 coins and saves 2 more every week. Ben has only 2 coins but saves 4 every week. Ben is catching up. In which week will they have exactly the same, and how could you find it without counting week by week forever?
2Explore
This activity needs a bigger screen
Turn your phone sideways, or open this lesson on a tablet or computer. The explanation and quiz below work here.
- Drag P onto the blue line. What happens to the blue check? Now put it on the red line.
- Turn off Show the crossing and hunt for the one point that gets two ✓.
- Make both slopes the same. Can the lines still meet?
- Make both lines exactly the same. How many points get two ✓ now?
3Explain
Two rules, one question
Ana has 10 + 2 × (weeks) coins; Ben has 2 + 4 × (weeks). A table shows they both have 18 in week 4. Two equations that must be true at the same time are called a system of equations.
A solution of a system makes every equation true at once. On a graph, it is the point where the lines cross. Lines cross once (one solution), never (parallel: none), or everywhere (same line: infinitely many).
4Practice
On a graph, the solution of a system of two equations is…
5Recap
- A system is two (or more) equations that must be true at the same time.
- Its solution is on every line: where the lines cross.
- To solve without a graph, set the two expressions for y equal and solve for x.
- Parallel lines: no solution. The same line twice: infinitely many.