Math101 · Independent Ontario learning library
Math101learn.math101.caLinear Equation
Solve a linear equation by preserving equality while undoing the operations around the variable.
A linear equation states that two expressions are equal and contains variables only to the first power.
Equality is a balance
An equation such as
is balanced for one particular value of $x$. Solving means finding that value without breaking the equality.
Inverse operations
| Operation present | Operation that undoes it |
|---|---|
| add $a$ | subtract $a$ |
| subtract $a$ | add $a$ |
| multiply by $a$ | divide by $a$ |
| divide by $a$ | multiply by $a$ |
A two-step equation
Why order matters
In $3x-7$, the value $x$ is multiplied by $3$ and then $7$ is subtracted. Solving reverses the process: undo subtraction first, then multiplication. It is like removing an outer layer before an inner layer.
Variables on both sides
Collecting variable terms on the side that leaves a positive coefficient is often convenient, but either side can work.
Equations with brackets
Fractions in an equation
When several fractions appear, multiplying every term by the least common denominator can clear them in one step.
One solution, none, or infinitely many
Most linear equations have one solution. Simplification can reveal two other cases.
These outcomes are information, not failed calculations.
Common mistakes
A solving checklist
- Simplify each side.
- Collect variable terms on one side.
- Collect constants on the other.
- Divide by the remaining coefficient.
- Check in the original equation.
Related topics
Explore the idea
Equation balance
Change one quantity at a time and connect what moves to Linear Equation.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve 3x − 7 = 11.
- 3x − 7 = 11
- 3x = 18
- x = 6
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
