Math101 · Independent Ontario learning library
Math101learn.math101.caEquation
An equation is a mathematical statement that two expressions have equal value. The equals sign expresses a relationship, not a command to calculate.
Equations encode constraints in science, finance, geometry, and functions. Interpreting equality relationally supports reliable algebra and prevents many invalid transformations.
Intuition and core definition
An equation is a mathematical statement that two expressions have equal value. The equals sign expresses a relationship, not a command to calculate. An equation may be true for all allowed values, true only for particular solutions, or false for all values. Solving means finding every allowed value that makes the statement true.
Notation, language, and conditions
The left-hand side and right-hand side are separated by $=$. A solution set lists all satisfying values. An identity such as $2(x+1)=2x+2$ holds for every real $x$; a conditional equation such as $2x+1=9$ holds for $x=4$; a contradiction such as $x+1=x+3$ has no solution.
Why this idea matters
An equation asserts that two expressions have the same value, and solving finds precisely the inputs that make that assertion true.
A dependable method
- Identify the domain and any restrictions.
- Simplify each side without changing the equality.
- Use inverse operations on both sides to isolate the variable.
- List all candidates, especially when operations such as squaring can introduce extras.
- Substitute each candidate into the original equation and compare both sides.
Worked example
Representations and interpretation
A balance scale models equality: adding, subtracting, multiplying, or dividing both sides by the same permissible amount preserves balance. A graph represents solutions as intersection points where the two side-expressions have equal outputs.
Reasoning about variations
Multiplying both sides by zero destroys information, and dividing by an expression that might be zero can lose solutions. Equivalent transformations require conditions; the balance metaphor includes only operations that can be reversed on the stated domain.
Common mistakes
How to check your work
- Substitute every solution into both original sides.
- Graph the two expressions and locate their intersections.
- Reverse each inverse operation to see whether any condition was lost.
Practice
- Solve $3x-5=16$.
- Classify $5(x+2)=5x+10$.
- Does $x=2$ solve $x^2+1=6$?
Answers and brief solutions
Show answers
- $x=7$ Add $5$ to both sides to get $3x=21$, then divide by $3$.
- Identity Distribution makes both sides equal for every real $x$.
- No The left side is $5$, not $6$.
Synthesis and transfer
A mobile-balance diagram can model an equation: removing equal weights from both sides preserves balance, while an unequal move changes the solution set.
Balance does not require the two sides to look alike; it requires them to evaluate equally for a particular input. The equation $3x+2=3x+2$ remains true for every real $x$, whereas $3x+2=3x+5$ becomes a contradiction after equal terms are removed. Those cases represent an identity and an equation with no solution, not failures of the solving process. A conditional equation such as $3x+2=14$ has only the input that preserves equality. Substitution into the original statement distinguishes all three outcomes and catches steps, such as multiplying by zero or squaring, that may change a solution set. Solving is therefore analysis of truth conditions, not simply moving symbols across a sign.
Related topics
Teaching and accessibility note
Explore the idea
Equation balance
Change one quantity at a time and connect what moves to Equation.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve $3x-5=16$.
- Add $5$ to both sides to get $3x=21$, then divide by $3$.
End of lesson
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