Solve Systems of Equations by Graphing
A system of equations is two equations that share the same two variables — something like and . A solution of the system is one ordered pair that makes both equations true at the same time. Not one or the other — both.
Graphing makes that idea visible. Every point on a line makes that line's equation true, so if you graph both equations, the one point where the lines cross is the one pair of numbers that works in both. Find the crossing point and you've solved the system.
What the solution of a system means
Each equation in a system has infinitely many solutions on its own — every point on its line. The system's solution is stricter: it has to satisfy both equations at once. A point that sits on only one of the two lines is not a solution of the system, no matter how nice it looks.
Write the solution as an ordered pair, -coordinate first: means and . Reversing the order gives a different point, so read carefully off the graph.
How to solve by graphing
Step 1: graph the first equation. If it's in slope-intercept form , start at the -intercept and use the slope to plot a second point, then draw the line.
Step 2: graph the second equation on the same axes, the same way.
Step 3: find the point where the two lines cross and read its coordinates — first, then . That ordered pair is the solution of the system. The graph below shows and crossing at .
Always check the point in both equations
A graph is only as accurate as your drawing, so confirm the answer with substitution. Plug the coordinates of the crossing point into both original equations. If both come out true, you're done. If one fails, the lines were drawn slightly off — regraph and look again.
This check is also the whole test for "is this point a solution?" questions: substitute the pair into each equation. Both true means yes; anything less means no.
Worked examples
Example 1: solve a system by graphing
Solve the system and by graphing.
Answer:
Example 2: a steeper line
Solve the system and by graphing.
Answer:
Example 3: test a point without graphing
Is a solution of the system and ?
Answer: Yes — is a solution of the system.
Try one yourself
Common questions
What if the lines cross at a point that isn't on nice grid lines?
Graphing gives you an estimate, and on messy numbers an estimate is all it gives. Read the closest grid point, then check it by substitution. If it doesn't check out exactly, the true solution is a fraction or decimal — algebra methods you'll learn later (substitution and elimination) find it exactly.
What if the two lines never cross?
Then the system has no solution — the lines are parallel, so no single pair works in both equations. Counting solutions is its own skill; see the article on how many solutions a system has.
Does it matter which line I graph first?
No. The crossing point is the same either way. Graph whichever equation looks easier first, and use a different color or pencil weight for each line so you can tell them apart.
How do I write the answer?
As an ordered pair: . If the lines cross where and , the solution is . You can also state it as and — same answer, different notation.
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