Math · Interactive
Systems of Equations
Two linear equations are two lines, and the solution to the system is where they cross. Drag both lines below to move the intersection and see one solution, no solution, or infinitely many. Free to use, and exportable into your slides.
Drag each line's slope and intercept to move the solution. Open fullscreen ↗
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What is a system of equations?
A system of equations is two or more equations that share the same variables, and solving the system means finding the values that make every equation true at once. For two linear equations, each equation is a straight line on the coordinate plane, and the solution is the point where the lines meet. Drag the two lines in the grapher above and the intersection point, the solution, moves with them.
Solving a system by graphing
To solve a system by graphing, plot both lines on the same axes and look for the point where they cross. The coordinates of that crossing point are the x and y that satisfy both equations. Graphing gives an immediate visual answer and shows why a system can have one solution, none, or infinitely many, though for exact non-integer answers you often confirm the point with algebra.
One solution, no solution, or infinitely many
Two lines with different slopes cross at exactly one point, so the system has one solution. Two lines with the same slope but different y-intercepts are parallel and never cross, so the system has no solution. Two lines that are actually the same line overlap everywhere, so the system has infinitely many solutions. The grapher labels which of these three cases you are looking at as you drag.
Checking the solution algebraically
Once graphing gives you a candidate point, substitute its coordinates back into both original equations to confirm they hold. You can also solve the system with substitution or elimination and compare: setting the two expressions for y equal gives x = (b₂ − b₁) / (m₁ − m₂), and substituting that x into either equation gives y. Matching the graph and the algebra is a good way to catch mistakes.
Frequently asked questions
What is a system of equations?
A system of equations is a set of equations with the same variables, solved together. For two linear equations the solution is the point where their graphs, two straight lines, intersect.
How do you solve a system of equations by graphing?
Graph both lines on the same coordinate plane and find where they cross. The coordinates of the intersection point are the solution that satisfies both equations at once.
When does a system have no solution?
A linear system has no solution when the two lines are parallel: they have the same slope but different y-intercepts, so they never meet. This is called an inconsistent system.
When does a system have infinitely many solutions?
When the two equations describe the same line (same slope and same y-intercept), the lines overlap completely, so every point on the line is a solution. This is a dependent system.
How do you find the intersection point algebraically?
Set the two expressions for y equal and solve for x, which gives x = (b₂ − b₁) / (m₁ − m₂), then substitute that x into either equation to get y. That (x, y) is the solution.
How do I use this systems of equations grapher?
Drag the slope and intercept sliders for each line. The two lines redraw instantly, the intersection is marked, and the panel shows the solution and whether the system has one, none, or infinitely many solutions. You can also export the graph.