Below is information about 5 linear systems represented in different ways. The first is the Sally/Arnie race. Fill-out the table for the next four (you are encouraged to use Desmos.com). Show how you solved the problem using each representation. Student Worksheet (Your work should also show connections between representations). Graphs are created using Desmos.com.
Solutions to linear systems can be shown for each representation as follows:
Equations | Substitution (or linear combinations – not highlighted in this module) |
Graphs | Intersection of lines (exact solution not always possible) |
Table | Same points in both tables (exact solution not always possible/often very tedious to find) |
Situation | Examination (exact solution not always possible) |
Three situations
Parallel |
Intersecting |
Coincident |
\(y=2x+4\) \(y=2x-1\) Soving… \(2x+4=2x-1\) \(4\neq-1\) (never true, no matter the value of \(x\)) |
\(y=-\Large \frac{1}{3}\normalsize x+2\) \(y=3x+2\) Solving… \(-\Large \frac{1}{3}\normalsize x+2=3x+2\) \(0=\Large \frac{10}{3} \normalsize x\) \(x=0\) \(y=2\) \((0,2)\) (there is only one \((x,y)\) point of intersection) |
\(y=2x+4\) \(4x-2y=-8\) Solving… \(4x-2(2x+4)=-8\) \(4x-4x-8=-8\) \(-8=-8\) (no matter the value of \(x\), always true) |
No solutions | One solution | Infinite solutions |