Solving simultaneous equations by substitution
Use this page when one equation already gives you a variable directly, or can be rearranged quickly, so substitution is the cleaner route.
Start here if you want the short version before reading the full method.
- Substitution works by rewriting one variable in terms of the other and placing that expression into the second equation.
- It is often the best route when one line already looks like y = ... or x = ....
What this topic means and what to look for first.
Substitution is often cleaner than elimination when one variable is already isolated, because you can move straight into a one-variable equation.
The main danger is forgetting brackets after substitution or expanding the new expression incorrectly.
One reliable route through the topic.
- 1Choose the equation that isolates a variable most easily.
- 2Rewrite that variable clearly in terms of the other one.
- 3Substitute the new expression into the second equation.
- 4Solve the resulting one-variable equation.
- 5Substitute back to find the second variable and then check both equations.
See the method in action.
y = 2x + 1 and x + y = 10
- Substitute y = 2x + 1 into x + y = 10.
- This gives x + 2x + 1 = 10, so 3x = 9 and x = 3.
- Substitute back to get y = 7.
x + y = 8 and 2x - y = 1
- Rearrange the first equation to y = 8 - x.
- Substitute into 2x - y = 1 to get 2x - (8 - x) = 1.
- Simplify to 3x - 8 = 1, so x = 3 and y = 5.
Things that commonly send the method off track.
- Substituting without brackets when the inserted expression contains more than one term.
- Rearranging the first equation incorrectly before substitution even begins.
- Solving for the first variable correctly but forgetting to find the second one.
Use a short verification pass before moving on.
- Use brackets around substituted expressions whenever there is more than one term.
- Check both final values in both original equations to make sure the system works as a pair.
Try a few variations before switching to a calculator or solver tool.
- y = x + 3 and 2x + y = 12
- x + y = 7 and 3x - y = 5
- y = 4x - 1 and x + y = 9
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Short answers worth checking.
It is often better when one variable is already isolated or can be rearranged quickly without creating messy fractions.
Brackets preserve the whole substituted expression so the signs stay correct when you simplify.
Put both final values into both original equations and confirm both equations still work.
Continue with the next closely related topic.
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