Math101 · Independent Ontario learning library
Math101learn.math101.caSolving One-Step Equations
A one-step equation isolates its variable with one inverse operation. The goal is not to “move” a term mysteriously but to preserve equality by applying the same legal operation to both sides.
One-step equations establish the balance principle and inverse-operation reasoning used in every later algebraic equation and formula.
Intuition and core definition
A one-step equation isolates its variable with one inverse operation. The goal is not to “move” a term mysteriously but to preserve equality by applying the same legal operation to both sides. Addition and subtraction are inverse operations; multiplication and nonzero division are inverses.
Notation, language, and conditions
Equations may appear as $x+a=b$, $x-a=b$, $ax=b$, or $x/a=b$ with $a\ne0$ where needed. The solution set contains values that satisfy the original. A coefficient includes its sign, so $-3x=12$ requires division by $-3$.
Why this idea matters
A one-step equation is undone with the inverse operation applied equally to both sides, preserving the original equality.
A dependable method
- Identify the operation directly attached to the variable.
- Choose its inverse operation.
- Apply that inverse to both sides and simplify.
- Write the isolated variable and solution clearly.
- Substitute the value into the original equation to verify equality.
Worked example
Representations and interpretation
A balance scale shows equal quantities on two pans. Removing the same weight, adding the same amount, or dividing equal groups on both sides preserves balance. A number-line model can show inverse translations for addition equations.
Reasoning about variations
For $x+0=9$, the identity addition changes nothing and $x=9$. For $0x=7$, no value works; division by zero is not an allowed inverse operation. Checking the coefficient prevents a false one-step routine.
Common mistakes
How to check your work
- Substitute into the untouched original equation.
- Reverse the inverse operation to reconstruct the given side.
- Estimate sign and magnitude before solving.
Practice
- Solve $x-13=-5$.
- Solve $-7x=35$.
- Solve $\frac{x}{6}=-4$.
Answers and brief solutions
Show answers
- $x=8$ Add $13$ to both sides.
- $x=-5$ Divide both sides by $-7$.
- $x=-24$ Multiply both sides by $6$.
Synthesis and transfer
A shared bill after a fixed discount can be modelled by one multiplication equation; reversing that operation and substituting back checks the per-person amount.
If a discounted total is $48$ dollars and four people share it equally, $4p=48$ is undone by dividing both sides by four. The operation is valid because the same nonzero scale is removed from both sides, yielding $p=12$. Multiplying the proposed share back by four checks the original relationship and restores the unit of dollars. A one-step equation may instead require addition, subtraction, or multiplication by a reciprocal; the inverse depends on the operation attached to the unknown. Writing the equality after every transformation emphasizes that a solution is preserved, rather than presenting the answer as the result of moving a number without justification.
Related topics
Teaching and accessibility note
Explore the idea
Equation balance
Change one quantity at a time and connect what moves to Solving One-Step Equations.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve $x-13=-5$.
- Add $13$ to both sides.
End of lesson
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