11. Systems from Tables and Graphs

A system may appear as a table, graph, or verbal description rather than two equations.

From a table, create a linear model by finding the starting value and rate of change. For points:
\[ (0,10),(1,15),(2,20), \]
the starting value is 10 and the rate of change is 5, so the model is:
\[ y=5x+10. \]

If another model is \(y=25\), the intersection solves:
\[ 5x+10=25. \]
Then \(5x=15\), so \(x=3\), and \(y=25\).

Graphs show the solution visually. Tables show it numerically. Equations show it symbolically. All three forms can describe the same system.

Common misconception: thinking a system must be written as two equations at the start. GED® problems often require you to build the equations first.

Academic habit: translate every representation into "starting value plus rate times input" when the relationship is linear.