Solving Systems of Equations Graphically
Solving a system graphically rests on one idea: a solution is a point lying on every curve at once, so the solutions are exactly the intersections. That makes the number of solutions visible before you do any algebra.
Why intersections are the solutions
Each equation in a system describes a set of points. The graph of
y = 2x + 1 is every point satisfying it. A solution to the
system must satisfy both equations, so it must lie on both graphs, which
means it sits where they cross.
That is the whole method. Graph everything, find the crossings, and read off the coordinates.
Check it: 2(2) + 1 = 5 and -(2) + 7 = 5. Both
hold, so (2, 5) is correct.
Two lines: three possible outcomes
One solution: the lines cross
Different slopes guarantee exactly one crossing. The system is called consistent and independent.
No solution: the lines are parallel
Same slope, different intercept. They never meet, so there is no solution
and the system is inconsistent. Solving algebraically
produces a contradiction such as 0 = 5, which is the algebraic
signature of parallel lines.
Infinitely many: the lines coincide
The two equations describe the same line, usually because one is a multiple
of the other. 2x + 4y = 8 and x + 2y = 4 are the
same line written twice. Every point on it solves the system, so there are
infinitely many solutions, and the system is dependent.
Algebraically you arrive at something always true, such as
0 = 0.
On a graph, the second line is drawn exactly on top of the first, so only one is visible. If you expected two lines and see one, check whether they are the same equation in disguise.
Getting a precise answer
Reading a crossing by eye is fine when the answer is a whole number. It fails as soon as it is not.
- Enter both equations, one per line.
- Make sure the crossing is inside the visible window.
- Open the Analyse tab and press the button.
- Every intersection is listed, labelled with which two expressions produced it, with a copy button.
Behind that button the calculator subtracts one function from the other and finds where the difference is zero, refining by bisection until it runs out of double precision. That is around six significant figures of reliable accuracy — far beyond anything you could measure off the picture.
Non-linear systems
Once curves are involved, the count of solutions stops being limited to zero, one or infinity.
A line and a parabola
Up to two intersections. y = x² and y = x + 2 meet
at (-1, 1) and (2, 4).
A line and a circle
Two intersections if the line cuts through, one if it is tangent, none if it misses. Graphing tells you which case you are in before you start the algebra, which is worth knowing because a tangent case has a repeated root and is easy to mishandle.
Two curves with no closed-form solution
y = ln(x) and y = 3 - x cross exactly once, and
that crossing cannot be written in elementary functions. There is no
algebraic route. Numerically, it is at about
x = 2.20794. This is the situation where graphical methods stop
being a convenience and become the only option.
Graphical or algebraic?
| Use graphing when | Use algebra when |
|---|---|
| You want to see how many solutions exist | You need an exact value such as a fraction or surd |
| The equations resist rearrangement | The question says "solve algebraically" |
| A decimal answer is acceptable | You must show working for marks |
| You want to sanity-check an algebraic answer | The solution involves values off the visible window |
Best practice is to use both. Solve algebraically for the exact answer, then graph it to confirm the crossing is where you claimed. A sign error that survives three lines of algebra is obvious the moment you see the picture.
Questions
How many solutions can a system of two equations have?
For two lines: one if they cross, none if they are parallel, infinitely many if they are the same line. Non-linear systems can have any number — a line and a circle can meet at two points, one, or none at all.
When should I solve graphically instead of algebraically?
Graphing is best when the equations are hard to manipulate, when you want to see how many solutions exist before committing, or when an exact answer is not required. Algebra is best when you need an exact value such as a fraction or a surd.
What does it mean if the lines are parallel?
The system is inconsistent and has no solution. Algebraically you reach a contradiction such as 0 = 5. The lines have the same slope and different intercepts, so they never meet.
What if both equations describe the same line?
The system is dependent and has infinitely many solutions: every point on the line satisfies both. Algebraically you reach something always true, such as 0 = 0.
How accurate is a graphical solution?
Reading a printed graph by eye gives about one decimal place. Using the intersection finder gives around six significant figures, because it solves numerically rather than measuring pixels.