Systems by Graphing
The solution of a system is where the lines cross.
About this topic
A system of equations is a set of two or more equations sharing the same variables. Graphing each line on the same axes shows the solution as the point where they cross, since that point satisfies both equations.
There are three possible outcomes: one solution when the lines intersect, no solution when they are parallel (same slope, different intercepts), and infinitely many solutions when the equations describe the same line. For example y = x + 1 and y = −x + 5 cross at (2, 3), and checking in both equations confirms it: 3 = 2 + 1 and 3 = −2 + 5.
Systems model any situation where two conditions hold at once, such as comparing two phone plans to see when they cost the same, or finding when two moving objects meet. The crossing point is the break-even point.
Graphing is visual and intuitive, but it can be imprecise when the intersection has fractional coordinates. Always check your answer by substituting into both original equations, and use algebraic methods for exact answers.
Key ideas
- Graph both lines
- Intersection gives (x, y)
- Parallel lines give no solution
Quick check
Use cases: where Systems by Graphing helps
- Homework: review the key idea behind Systems by Graphing before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Systems by Graphing without notes.
- Visual learning: play with the interactive lab to see how Systems by Graphing changes when you adjust its inputs.
- Teaching a friend: explain Systems by Graphing in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Systems by Graphing?
The solution of a system is where the lines cross.
What will I learn in Systems by Graphing?
Graph both lines. Intersection gives (x, y). Parallel lines give no solution.
Is Systems by Graphing part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Systems by Graphing lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Systems by Graphing?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.