📚 Practice With Worksheets
Practice solving and graphing systems of linear equations with the resources below. The 🛠️ icon identifies a companion worksheet designed specifically for use with the interactive tool above.
▶️ Learn With Videos
Review how to solve systems by graphing, substitution, and elimination while connecting each algebraic method to the intersection of two lines.
🛠️ How to Use This Tool
Adjust the slope and y-intercept sliders for Equation 1 and Equation 2. Watch how the two lines move and how the number of solutions changes.
Select One Solution, No Solution, or Infinite Solutions to generate and explore a system with that classification.
Use the Substitution Solution and Elimination Solution tabs to compare two algebraic methods for solving the same system.
🧠 Things to Notice
When two lines have different slopes, they intersect once. The coordinates of that intersection satisfy both equations.
When two lines have the same slope but different y-intercepts, they are parallel and never intersect.
When two equations represent the same line, every point on that line satisfies both equations, so the system has infinitely many solutions.
📘 Vocabulary
System of linear equations: Two or more linear equations considered together.
Solution of a system: An ordered pair that makes every equation in the system true.
Intersection point: The point where two lines cross. Its coordinates are the solution to a one-solution system.
Substitution: A method in which one expression is substituted for an equivalent variable or expression in another equation.
Elimination: A method in which equations are added or subtracted so that one variable is removed.
One solution: A system whose lines intersect at exactly one point.
No solution: A system whose lines are parallel and never intersect.
Infinitely many solutions: A system whose equations represent the same line.
Consistent system: A system that has at least one solution.
Inconsistent system: A system that has no solution.
Independent system: A system with exactly one solution.
Dependent system: A system in which both equations represent the same line.
📐 Understanding the Number of Solutions
One solution: The two lines intersect at exactly one point. The coordinates of that point make both equations true. This normally occurs when the two lines have different slopes.
\(\text{Different slopes} \longrightarrow \text{one intersection point} \longrightarrow \text{one solution}\)
No solution: The two lines have the same slope but different y-intercepts. The lines are parallel, so there is no ordered pair that belongs to both lines.
\(\text{Same slope, different intercepts} \longrightarrow \text{parallel lines} \longrightarrow \text{no solution}\)
Infinitely many solutions: The equations are equivalent and graph as the same line. Every point on the line satisfies both equations.
\(\text{Same slope, same intercept} \longrightarrow \text{same line} \longrightarrow \text{infinitely many solutions}\)
Algebraically, a one-solution system produces values for both variables. A no-solution system simplifies to a false statement, such as \(0=5\). An infinite-solution system simplifies to a true identity, such as \(0=0\).
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