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Solution in Math — Definition, Meaning & Examples

Solution
Solution Set

Any and all value(s) of the variable(s) that satisfies an equation, inequality, system of equations, or system of inequalities.

With a system of equations or system of inequalities, the solution set is the set containing value(s) of the variable(s) that satisfy all equations and/or inequalities in the system.

Example 1: The only solution to the equation a−4=1a-4=1 is a=5.a=5\text{.}

Example 2: One possible solution to the inequality a−4≤1a-4\le 1 is a=3.a=3\text{.} The set of all solutions is {a:a≤5}\{a : a\le 5\} or (−∞,5].(-\infty,5]\text{.}

Example 3: The only possible solution to the system {x+y=5x−y=3\begin{cases} x+y=5 \\ x-y=3 \end{cases} is (x,y)=(4,1).(x,y)=(4,1)\text{.}

Example 4: One possible solution to the equation x+y=5x+y=5 is (x,y)=(2,3).(x,y)=(2,3)\text{.} The set of all solutions is {(x,y):y=5−x and x∈ℜ}.\{(x,y) : y=5-x \text{ and } x\in\Re\}\text{.}

See also

Set-builder notation, interval notation, equivalent system of equations, verify a solution

Key Formula

If f(x)=0, then x=a is a solution when f(a)=0.\text{If } f(x) = 0, \text{ then } x = a \text{ is a solution when } f(a) = 0.
Where:
  • f(x)f(x) = An expression or function involving the variable x
  • xx = The unknown variable you are solving for
  • aa = A specific value that, when substituted for x, makes the equation true

Worked Example

Problem: Find the solution of the equation 2x + 6 = 14.
Step 1: Subtract 6 from both sides to isolate the term with x.
2x+6−6=14−62x + 6 - 6 = 14 - 6
Step 2: Simplify both sides.
2x=82x = 8
Step 3: Divide both sides by 2.
x=82=4x = \frac{8}{2} = 4
Step 4: Verify by substituting x = 4 back into the original equation.
2(4)+6=8+6=14  ✓2(4) + 6 = 8 + 6 = 14 \; \checkmark
Answer: The solution is x = 4.

Another Example

This example involves a system of two equations with two unknowns, showing that a solution must satisfy every equation in the system simultaneously.

Problem: Find the solution set of the system of equations: x + y = 10 and 2x − y = 2.
Step 1: Add the two equations together to eliminate y.
(x+y)+(2x−y)=10+2(x + y) + (2x - y) = 10 + 2
Step 2: Simplify. The y terms cancel.
3x=123x = 12
Step 3: Solve for x by dividing both sides by 3.
x=4x = 4
Step 4: Substitute x = 4 into the first equation to find y.
4+y=10  ⟹  y=64 + y = 10 \implies y = 6
Step 5: Verify in the second equation.
2(4)−6=8−6=2  ✓2(4) - 6 = 8 - 6 = 2 \; \checkmark
Answer: The solution is (x, y) = (4, 6).

Frequently Asked Questions

What is the difference between a solution and a solution set?
A solution is a single value (or ordered pair, triple, etc.) that satisfies a mathematical statement. A solution set is the collection of all solutions. For example, the equation x² = 9 has two solutions, x = 3 and x = −3, so its solution set is {3, −3}.
Can an equation have no solution?
Yes. An equation with no solution is called a contradiction. For instance, x + 2 = x + 5 simplifies to 2 = 5, which is never true. No value of x can make this equation hold, so the solution set is the empty set, written ∅ or { }.
How do you verify a solution is correct?
Substitute the value back into the original equation or inequality. If the left side equals the right side (for an equation) or the inequality holds true, then the value is a valid solution. Always check using the original statement, not a simplified version, to avoid errors introduced during solving.

Solution of an Equation vs. Solution of an Inequality

Solution of an EquationSolution of an Inequality
DefinitionA value that makes two expressions equalA value that makes one expression greater than, less than, or equal to another
Number of solutionsOften a finite number (e.g., 1 or 2 values)Usually infinitely many values forming a range or region
How to expressSpecific values like x = 4 or a set like {3, −3}Interval notation like x > 2, or (2, ∞), or a shaded region on a graph
Graphical meaningPoints where a curve crosses a line or axisRegions above, below, or between curves

Why It Matters

Finding solutions is the central goal of algebra — nearly every algebra problem asks you to solve an equation, inequality, or system. Solutions also appear throughout geometry (solving for unknown angles or lengths), science (finding when a projectile lands), and finance (determining break-even points). Understanding what a solution is and how to verify one is essential for every math course from pre-algebra through calculus.

Common Mistakes

Mistake: Forgetting to check for extraneous solutions, especially with rational equations or equations involving square roots.
Correction: Always substitute your answers back into the original equation. Operations like squaring both sides or multiplying by a variable expression can introduce values that don't actually satisfy the original equation.
Mistake: Reporting only one solution when an equation has multiple solutions (e.g., writing x = 5 for x² = 25 and missing x = −5).
Correction: Consider all possible cases. For quadratics and higher-degree equations, factor completely or use the quadratic formula to find every solution. State the full solution set.

Related Terms