Math101 · Independent Ontario learning library
Math101Equation
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.
