Systems of Equations
Solve two linear equations by elimination — every step explained, with exact fraction answers.
- 1
The system of equations.
2x + 3y = 12
x − y = 1
- 2
Line up the y-terms: multiply equation 1 by b₂ and equation 2 by b₁.
-2x − 3y = -12
3x − 3y = 3
- 3
Subtract the equations to eliminate y, then solve for x.
-5x = -15
x = 3
- 4
Substitute x back in to find y.
2x + 3y = 12 with x = 3
y = 2
x = 3, y = 2
About the Systems of Equations Solver
Solving a system of two linear equations by elimination involves a handful of moves that students can each do correctly on their own and still mess up the order of. This tool walks through the actual elimination process step by step: lining up the y-coefficients, subtracting to cancel a variable, solving for x, then substituting back to find y, with a plain-language reason attached to each move.
What I like about building it around elimination specifically is that it forces students to see the multiplication step that creates matching coefficients, rather than jumping straight to a memorized formula. The reasoning toggle on each step exists for the kid who solved correctly but couldn't tell you why it worked.
How to use it in your classroom
- Enter the coefficients a, b, and c for both equations in the form ax + by = c.
- Read the system as written in the first step card, confirming both equations are entered correctly.
- Step through the elimination process: lining up coefficients, eliminating one variable, solving for the first variable, then substituting to find the second.
- Tap "Show reasoning" on any step to see why that particular move was made.
- Check the final answer panel for the solved values of x and y.
Tips from the classroom
- Start with a system where one equation already has no y-term, like x = 5 paired with another equation — the tool skips the coefficient-matching step and shows students what elimination looks like in its simplest form.
- Set up a system with parallel lines (matching coefficients but different constants) and let students discover what "no solution" looks like in the steps before you explain it.
- Use a system with coincident lines to introduce "infinitely many solutions," since most algebra students have never seen that outcome demonstrated step by step.
- Have students predict the solution by graphing the two lines on paper first, then check their intersection point against this tool's algebraic answer.
Frequently asked questions
What does the tool do when the two equations represent parallel lines?
It checks the coefficients, finds no valid elimination, and reports either "no solution" if the lines never meet or "infinitely many solutions" if the two equations actually describe the same line.
Does the tool show the multiplication needed to line up coefficients?
Yes, when neither equation already isolates a variable, it displays the multiplied versions of both equations side by side before showing the subtraction step that eliminates y.
Can I enter negative coefficients or constants?
Yes, all six values — a, b, and c for each equation — accept negative numbers, and the equation display formats negative terms correctly with proper signs.