Mathematics
Systems of Equations
A system of equations is a set of equations sharing the same variables. The solution is the set of values that satisfy every equation at once. Three main methods: substitution, elimination, matrix row-reduction.
How to Approach Systems of Equations
Choose the method
Substitution: pick the easiest equation to solve for one variable. Elimination: align variables and add/subtract to cancel one. Matrix: write the augmented matrix and row-reduce.
Reduce to one variable
After applying the method, you'll have a single equation in one unknown. Solve it.
Back-substitute
Plug the value back into one of the original equations to find the remaining variables. Check by substituting into all original equations.
Frequently Asked Questions
When do I use elimination vs substitution?+
Elimination is faster when the coefficients are integers and one variable's coefficient matches across equations. Substitution is faster when one equation is already solved for a variable.
What if there's no solution?+
Geometrically, parallel lines never intersect. Algebraically, you'll derive a false statement like 0 = 7. The system is 'inconsistent'.
What about three or more variables?+
The matrix method (Gaussian elimination) scales cleanly. ScanSolve handles 3x3 and larger systems with full row-reduction shown.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
Stuck on a Systems of Equations problem?
Snap a photo or type the question. ScanSolve walks you through every step — same as the worked examples above. 5 free solves per day, no card required.