Connecting a Table, an Equation, and a Graph in Context

A table of tutoring costs looks like a list of numbers. A graph of the same story looks like a tilted line. An equation looks like y = mx + b. They feel different because they show the information in different formats, but underneath they're tracking the exact same two quantities changing together: hours and dollars. Once you see that, you can move between all three freely, and on test day you often have to.

The key is reading what each axis means. If hours sit on the horizontal axis and dollars sit on the vertical axis, then one step right (one more hour) corresponds to some number of steps up (more dollars). That ratio, dollars per hour, is the slope in the equation and the rate of change in the table: divide the change in cost by the change in hours. The starting value, what you'd pay for zero hours, is the y-intercept in the equation and the 'base' row in the table if one exists.

This connects directly to systems and inequalities nearby in the algebra domain: once you can pull an equation out of a table or graph, comparing two linear situations becomes a system, and asking when one costs less than another becomes an inequality. A flat fee added to a rate is the same move as shifting a graph straight up without changing its tilt. Recognizing that move instantly saves you from rebuilding the whole equation from scratch.