2. Reading Systems from Graphs and Tables

Introduction

Before you solve a system with algebra, it helps to recognize what a solution looks like when
it is shown to you directly — on a graph, or hidden inside a table of \(x\)- and \(y\)-values.
The digital SAT® frequently gives you this information for free; this session trains you to
read it instead of re-deriving it.

Definition

On a graph, the solution of a two-variable linear system is the point where the two lines
intersect — read its coordinates directly from the axes. In a table of values, if two
different equations are each evaluated at the same \(x\), the solution is the \(x\) at which
both equations produce the same \(y\)-output.

Real-Life Uses

• A phone bill comparison table lists monthly costs for two carriers at 0, 5, 10, and 15 GB;
  the row where the totals match is the system's solution.
• A fitness app graphs two people's calories burned over time; the point where the curves cross
  shows when they have burned the same amount.
• Weather data tables list two rivers' water levels by day; a matching row signals equal levels.

Why You Should Know This

On test day, some systems questions supply only a picture or a table and never ask you to write
an equation at all. Recognizing the intersection point immediately — without unnecessary
algebra — saves time for harder questions elsewhere on the section.

Worked Examples

Example 1

Problem: A graph shows line \(A\) passing through \((0,1)\) and \((4,9)\), and line \(B\) passing through \((0,7)\) and \((4,3)\). The lines cross at \(x=2\). What is the \(y\)-value at the crossing point?

Solution: Line \(A\) has slope \(\frac{9-1}{4-0}=2\), so \(y=2x+1\). At \(x=2\), \(y=2(2)+1=5\). Line \(B\) has slope \(\frac{3-7}{4-0}=-1\), so \(y=-x+7\); at \(x=2\), \(y=5\). Both agree, confirming the solution \((2,5)\).

Example 2

Problem: A table shows: at \(x=0\), \(y_1=3\) and \(y_2=11\); at \(x=2\), \(y_1=7\) and \(y_2=7\); at \(x=4\), \(y_1=11\) and \(y_2=3\). What is the solution of the system?

Solution: Compare the two \(y\)-columns at each \(x\). At \(x=2\), both columns equal \(7\). The solution is \((2,7)\).

Example 3

Problem: A graph shows two lines that appear to cross near \(x=3\), one increasing steeply and one nearly flat. Why can you not be fully certain of the solution from the picture alone?

Solution: A graph read by eye only gives an approximate crossing location. Unless the axes are marked precisely at that point or the lines pass through clearly labeled lattice points, the exact solution should be confirmed algebraically.

Example 4

Problem: A table lists \(y=3x-2\) and \(y=-x+10\) for \(x=1,2,3,4\). Use the table to find where the two expressions are equal.

Solution: Compute both: at \(x=1\), \(1\) and \(9\); at \(x=2\), \(4\) and \(8\); at \(x=3\), \(7\) and \(7\) — equal; at \(x=4\), \(10\) and \(6\). The solution is \(x=3\), giving \(y=7\), so \((3,7)\).

Example 5

Problem: A graph of two lines shows they are parallel and never touch. What does this tell you about the system's solution from the picture alone?

Solution: Parallel lines that never intersect indicate the system has no solution — there is no ordered pair that lies on both lines simultaneously.

Example 6

Problem: A table for two equations shows the same pair of values, \((5,12)\), repeated at every listed \(x\) except one row, which differs by a small graphing error. What is the most likely explanation?

Solution: If nearly every row matches, the two equations are almost certainly the same line (infinitely many solutions), and the one differing row is a recording or rounding error rather than evidence of a different intersection.