Algebra
Systems of equations and inequalities
Solving systems by graphing
20 practice questions
2 video lessons
Theory + worked examples
Theory
To solve a system by graphing, graph both equations. The intersection point \((x,y)\) is the solution — it satisfies both.
Parallel lines give no solution; the same line gives infinitely many.
The lines cross at \((1,2)\), the solution.
Solving by graphing.
The solution:
\[\text{solution}=\text{intersection point}\]
Check the point in both equations.
How to solve by graphing
- Graph the first equation.
- Graph the second on the same axes.
- Find the intersection point.
- Check it in both equations.
Example 1 — Find the intersection
Solve \(y=x+1\) and \(y=-x+3\) by graphing.
Solution
The lines cross at one point.
| \(\text{intersection}\) | \(=\) | \((1,2)\) |
Example 2 — Check the solution
Check \((1,2)\) in both equations.
Solution
Substitute into each.
| \(2\) | \(=\) | \(1+1\ \checkmark\) |
| \(2\) | \(=\) | \(-1+3\ \checkmark\) |
Example 3 — Parallel lines
What if two lines are parallel?
Solution
They never meet — no solution.
Example 4 — Read a graph
Two graphs meet at \((3,5)\). Give the solution.
Solution
The intersection point is the solution.
| \((x,y)\) | \(=\) | \((3,5)\) |
Common pitfalls
Graph accurately to read the intersection.
The solution is a point \((x,y)\).
Parallel lines have no solution.
Frequently asked questions
How do you solve a system by graphing?
Graph both equations and find where they intersect.
What is the solution of a system?
The point that satisfies all equations.
What if the lines are parallel?
No solution.
Should you check the solution?
Yes — in both equations.
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