Two equations, two unknowns—use substitution or elimination to find where they intersect.
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Solve Now →A system is two (or more) equations that must be true at the same time. The solution is the values of x and y that satisfy both equations.
Graphically, it's where two lines intersect. Algebraically, we find x and y values that work in both equations.
For a visual explanation, see Purplemath's systems guide.
Best when: One variable is already isolated (like y = ... or x = ...)
Pick the equation where it's easiest to solve for x or y.
Replace that variable with the expression you found.
Now you have one equation with one unknown—solve it!
Plug your answer back into either equation to find the other variable.
Substitution Example
Solve: y = 2x + 1 and 3x + y = 11
Best when: Coefficients are opposites or can be made opposites easily
Write in standard form (Ax + By = C) with variables aligned.
Multiply one or both equations so one variable has opposite coefficients.
One variable cancels out! Solve for the remaining variable.
Plug your answer into either original equation.
Elimination Example
Solve: 2x + 3y = 12 and 4x - 3y = 6
| If you see... | Use... |
|---|---|
| One variable already isolated (y = ...) | Substitution |
| Coefficient of 1 or -1 (easy to isolate) | Substitution |
| Opposite coefficients already (like +3y and -3y) | Elimination |
| Same coefficients (like 2y and 2y) | Elimination (subtract) |
| No obvious choice | Either works! |
❌ No Solution
If you get something like 0 = 5 (false), the lines are parallel—they never intersect.
Example: x + y = 3 and x + y = 7
∞ Infinite Solutions
If you get 0 = 0 (always true), the equations are the same line.
Example: 2x + 4y = 8 and x + 2y = 4
Substituting into the wrong equation
After solving for x, plug into an ORIGINAL equation, not one you modified.
Forgetting to multiply ALL terms
When multiplying an equation by a number, multiply EVERY term including the constant.
Not checking in BOTH equations
Your answer must work in both original equations—always verify!
Need 3 variables? See our guide on solving systems with 3 variables.
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