Math101 · Independent Ontario learning library
Math101learn.math101.caSolving Systems by Graphing
Solving a linear system by graphing identifies the shared point or shared set of its lines.
The graphical solution of a system is every point where the graphs intersect.
Why intersection solves the system
A point on the first line satisfies its equation. A point on the second satisfies the other. At an intersection, the same ordered pair lies on both lines, so both equations are true at once.
Graphing makes the meaning of a system visible before algebraic methods compress the work.
A reliable graphing method
- Rewrite each equation in a graphable form such as $y=mx+b$ when useful.
- Choose an axis window and scale containing likely intersections.
- Plot at least two accurate points for each line.
- Draw the lines and read the intersection.
- Substitute the pair into both equations for an exact check.
Different colours and labels prevent the two equations from being mixed.
Worked example
The graph suggests the pair; substitution confirms it exactly.
Graphing standard form
For $2x+3y=12$, use intercepts or isolate $y$. Setting $x=0$ gives $(0,4)$ and setting $y=0$ gives $(6,0)$. Two intercepts determine the line.
If an intercept is inconvenient, choose input values that produce manageable outputs.
Exact versus approximate answers
An intersection such as $(4/3,11/6)$ may be difficult to read accurately from a hand graph. Report a graphical estimate with an approximation sign, then use substitution or elimination when an exact value is required.
Graphing technology can refine the estimate, but the selected viewing window and numerical rounding still matter.
No solution
Parallel distinct lines never meet. In slope-intercept form they have equal slopes and different intercepts, such as $y=3x+1$ and $y=3x-5$.
A graph may make nearly parallel lines look parallel in a small window. Compare slopes or zoom thoughtfully before concluding.
Infinitely many solutions
Equivalent equations graph as the same line. For example, $2x+4y=8$ and $x+2y=4$ overlap completely. Every point on that line solves both equations.
If only one line appears, test whether one equation is a constant multiple of the other rather than assuming a graphing error.
Context and feasible regions
A context may restrict the relevant part of a line. Negative ticket counts are not feasible even if the algebraic lines extend there. Plot and interpret the domain that matches the situation.
Common mistakes
Reading the crossing before plotting accurately. Use precise points and scales.
Reporting separate intercepts as the solution. The system solution is the lines’ shared point.
Using $=$ for an estimate. Use $\approx$ when the graph cannot support exactness.
Missing overlap. Simplify equations when graphs appear identical.
Quick self-check
- Are both axes labelled with a useful scale?
- Did I graph each equation, not just one?
- Is the intersection exact or estimated?
- Do both original equations accept the pair?
- Could the lines be parallel or coincident?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Where do the lines y = x + 2 and y = −2x + 8 intersect?
- Set x + 2 = −2x + 8.
- Then 3x = 6, so x = 2.
- Substitute to get y = 4.
End of lesson
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