Expressions & Equations

Solving Systems Algebraically

Solving a system of two linear equations by substitution and elimination.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Solving Systems Graphically

Finding an Exact Intersection Without Graphing

A system's intersection point doesn't always land on a nice, easy-to-read grid point — before reading on, think about how you might find an exact solution using just algebra, without needing to graph precisely enough to read off a messy coordinate.

Substitution replaces one variable with an equivalent expression from the other equation, collapsing two equations with two unknowns down into one equation with one unknown — genuinely solvable using the equation-solving skills already built up. It works because if two expressions both equal y, they must equal each other too.

Worked Example — Solving a System by Substitution

Solve y = x + 1 and y = 3x − 5. Since both expressions equal y, set them equal to each other: x + 1 = 3x − 5. Solve: 6 = 2x, so x = 3. Substitute back into either original equation: y = 3 + 1 = 4. The solution is (3, 4). Check in the other equation: 3(3) − 5 = 4. True.

Elimination adds or subtracts the two equations directly, chosen so that one variable cancels entirely — this works because adding equal quantities to both sides of a true equation, or adding two true equations together, always produces another true equation.

Worked Example — Solving a System by Elimination

Solve 3x + 2y = 16 and 3x − 2y = 8. Adding the two equations directly cancels the y terms: 6x = 24, so x = 4. Substitute into the first equation: 3(4) + 2y = 16, giving 2y = 4, so y = 2. The solution is (4, 2). Check in the second equation: 3(4) − 2(2) = 12 − 4 = 8. True.

Tip

Choose substitution when one equation is already solved for a variable (or easy to solve for one); choose elimination when the coefficients of one variable are already equal or opposite, or can easily be made so by multiplying an equation.

Common Mistakes

Forgetting to substitute the found value back in to solve for the second variable, leaving the solution incomplete.
A system's solution needs both coordinates — after solving for one variable, always substitute it back into an original equation to find the other.
Adding two equations in elimination when the target variable's coefficients have the same sign, causing it to double instead of cancel.
Check whether the coefficients being eliminated are opposites (add the equations) or identical (subtract the equations) before combining them.

Key Takeaways

Substitution replaces one variable with an equivalent expression, reducing a system to a single equation with one unknown.
Elimination adds or subtracts the two equations to cancel one variable entirely.
Both methods find the exact intersection point that graphing can only approximate.

Summary

Substitution and elimination solve systems exactly, using algebra built from earlier equation-solving skills. The next unit shifts from equations to the broader idea of a function — a rule connecting inputs to outputs.

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