What solving a pair means
Each equation is a straight line. A solution is the single point where the two lines cross, so it must satisfy both equations at once.
2x + 3y = 12 and x − y = 1 meet at x = 3, y = 2, because 6 + 6 = 12 and 3 − 2 = 1.
Worked example
Take 2x + 3y = 12 and x − y = 1. The determinant is 2 × (−1) − 1 × 3 = −5.
x = (12 × −1 − 1 × 3) / −5 = 3 and y = (2 × 1 − 1 × 12) / −5 = 2. Both checks come back as 12 and 1.
When there is no single answer
A determinant of zero means the lines are parallel or the same line. Parallel lines never meet, so there is no solution.
If the two equations are multiples of each other, every point on the line solves both, so there are infinitely many solutions.
Inputs for the simultaneous equations calculator
Equation 1
The coefficients of x and y, then the number on the right-hand side.
Equation 2
The same three numbers for the second equation, including their signs.
CALCULATOR QUESTIONS
Simultaneous equations calculator questions
How do I solve simultaneous equations?
Eliminate one variable by making its coefficients match, then subtract. Here the determinant method does the same work in one step.
What does the determinant tell me?
A non-zero determinant means the lines cross once. A zero determinant means they are parallel or identical, so there is no unique answer.
Why are both checks shown?
A solution has to satisfy both equations. If either check is wrong, the pair was copied in with a mistake.
Can I solve three unknowns?
Not on this page, which handles two equations in two unknowns. Three unknowns need three equations and a different method.