How to solve a system of linear equations, step by step

A system of linear equations is two or more equations that share the same unknowns. Solving it means finding the values that make every equation true at once. For two equations in $x$ and $y$, that pair is the point where the two lines cross.

Method 1: Substitution

Substitution works best when one equation already has a variable on its own, or is easy to rearrange into that form. You solve one equation for a variable, then put that expression into the other.

Method 2: Elimination

Elimination shines when the variables line up in columns. You add or subtract the equations so one variable cancels, then solve for the other. Scale an equation first if the coefficients do not match.

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Which method should you use?

• Substitution: pick it when a variable is already isolated or has a coefficient of $1$.
• Elimination: pick it when the coefficients line up or are easy to match by scaling.
• Both give the same answer, so use whichever needs less arithmetic for the numbers in front of you.

No solution or infinitely many

Not every system has one neat answer. If your steps collapse to something false like $0 = 5$, the lines are parallel and there is no solution. If they collapse to something always true like $0 = 0$, the two equations describe the same line and there are infinitely many solutions.

Check any solution by putting the pair back into both original equations. If both hold, you are done.