Same Math › Algebra 1 · grades 8–10 › Systems by substitution and elimination
How do you solve a system with substitution or elimination?
A system is two equations that share the same x and y. Graphing finds where the lines cross; substitution and elimination find it exactly. Substitution plugs one equation into the other. Elimination adds (or subtracts) the equations so one variable disappears. Either way, the answer is a point (x, y) that works in both.
- Substitution: best when one equation already says y = or x =.
- Elimination: best when a variable cancels when you add or subtract.
- Find one variable, then plug back in for the other.
Why schools teach it
Exact answers when graphs only estimate
Algebra 1 students solve systems exactly and understand why adding equations keeps the solution the same (HSA.REI.C.5 and C.6). Graphing, which many kids met in eighth grade, only works when the crossing point is on the grid.
Picking the right method is part of the skill: substitution when a variable is already alone, elimination when the numbers line up.
At the kitchen table
What to say, and what to skip
Say this
Is a variable already alone? Then substitute.
Which variable cancels if you add the equations?
Or multiply one first so it will.You found x. How do you get y?
Plug x into either equation.Does your point work in both equations?
Skip this
- Stopping after finding x.The answer is a point: (x, y).
- Multiplying only part of an equation.Every term gets multiplied.
- Forgetting parentheses when substituting.x + (2x + 1) = 7, not x + 2x + 1 = 7 with a lost sign.
See it in action
Solving 2x + 3y = 12 and x − 3y = −3
The y terms are 3y and −3y. Add the equations
3x = 9, so x = 3Plug x = 3 into either equation
3 − 3y = −3, so y = 2Write the point and check
(3, 2): 6 + 6 = 12 ✓
If nothing cancels, multiply one equation first, like ×2, so it does.
Free practice sheet
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Parents also ask
What if both variables cancel?
If you get something true like 0 = 0, there are infinitely many solutions (same line). Something false like 0 = 5 means no solution (parallel lines).
Which method is better?
Neither. Substitution when one variable is alone; elimination when coefficients match or can be made to match.
Why does adding equations work?
Adding equal amounts to both sides keeps an equation balanced, and the second equation says two things are equal.
Do teachers want a check?
Many do. Plug the point into both original equations.