Connect models, tables, graphs, and equations for linear relationships.
Tables, graphs, equations, and word problems can all encode the same linear rule.
Core Idea
A linear relationship appears the same way across representations: constant differences in a table, a straight line on a graph, and a first-degree equation. Moving between these forms is about extracting slope and a point.
Understanding
Tables, graphs, equations, and word descriptions are four windows into the same relationship. The SAT tests whether you can move between them fluidly.
From a table: check that y-values change by a constant amount for equal x-steps. That constant change is the slope. Pick any row to get a point, then build the equation.
From a graph: identify two clear points, compute slope, and read or compute the y-intercept.
From a word problem: identify the rate of change (slope) and the starting value (y-intercept). A phrase like "$12 per hour plus a $30 fee" translates directly to
The key skill is recognizing that all these forms encode the same two numbers: slope and y-intercept.
Step by Step
- From a table: compute
Δ 𝑦 Δ 𝑥 - From a graph: pick two lattice points, compute slope, and identify the y-intercept.
- From a word problem: identify the per-unit rate (slope) and the fixed/starting quantity (y-intercept).
- Write the equation in the form the question requests, or match to the correct graph/table.
Misconceptions
- Assuming a table is linear without checking — if the differences between y-values aren't constant for equal x-steps, the relationship isn't linear.
- Mixing up the independent and dependent variables when reading a word problem — the quantity that changes freely is x, the quantity that depends on it is y.
- Ignoring units in context problems — a slope of 12 means 12 dollars per hour, not just 12.
Worked Example
A table shows that when
Select an answer to see the explanation