A solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.
A solution can be one number, a whole range of numbers, a point on a graph, or nothing at all. In every case it is whatever makes the statement true.
Formal definition and notation
A solution of an equation or inequality in the variables x1, …, xn is an assignment of numbers to those variables that turns it into a true statement. The set of all solutions is the solution set. A linear equation in one variable has exactly one solution, no solution, or infinitely many; an equation in two variables has a graph that is exactly the set of its solutions.
Symbol
Read it as
Meaning
x = 4
x equals 4
a single solution of a one-variable equation
(2, 3)
the ordered pair 2, 3
a solution of an equation or system in two variables
{−2, 5}
the set negative 2, 5
a solution set with two solutions
∅
the empty set
no solution
Key facts
Solving means finding all the solutions. Checking means testing one value.
An inequality usually has many solutions: x > 7 has many solutions, every number greater than 7.
A linear equation has one solution, no solution (1 = 3 at the end) or infinitely many (0 = 0 at the end).
For two variables a solution is an ordered pair. Every point on the graph of y = 2x − 1 is a solution of that equation.
The solution of a system of equations must make every equation true, so it is where the graphs intersect.
How to check a solution
How to check a solution
substitute the value → simplify each side → compare
Substitute
replace every copy of the variable with the value, using parentheses
Simplify
work out each side separately with the order of operations
Compare
if the statement is true (=, < or > holds), the value is a solution; if not, it is not
Always check a solution by substituting it back into the original statement, not into a step you wrote later, because a copying mistake in a later step would also hide in the check.
Try it: find the solution
Interactive
Undo the operations on both sides until x is alone. The value you end with is the solution, and the readout checks it in the original equation.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
The equation is the statement, such as x + 2 = 6. The solution is the value that makes it true, x = 4.
Solutionvs.Zero (root) of a function
A zero of a function is an input that makes the output 0, such as x = 3 for f(x) = x − 3. It is the solution of the equation f(x) = 0, a special case of a solution.
Quick check: 3 questions
Question 1 of 3
Which value is the solution of 5x − 2 = 18?
Show answer and explanation
Answer: B. Add 2 to both sides: 5x = 20, so x = 4. Check: 5(4) − 2 = 18. Choice A subtracts 2 instead of adding it, C forgets to divide by 5, and D gives 5(3) − 2 = 13.
Question 2 of 3
Which ordered pair is a solution of 3x + 2y = 12?
Show answer and explanation
Answer: D.3(2) + 2(3) = 6 + 6 = 12. True. The others give 9 + 4 = 13, 12 + 2 = 14 and 0 + 8 = 8. Note that (3, 2) and (2, 3) are different points.
Question 3 of 3
How many solutions does 2(x + 3) = 2x + 6 have?
Show answer and explanation
Answer: C. Distribute: 2x + 6 = 2x + 6. Both sides are the same for every x, so every number is a solution. Choices A and D are each true values but not the only ones, and B would need a false statement such as 6 = 8.
Solution: frequently asked questions
Can an equation have no solution?
Yes. x + 1 = x + 3 has no solution: subtracting x leaves 1 = 3, which is false for every number. Some equations have no real solution for other reasons; x2 = −4 has none among real numbers, because no real number squared is negative. Write the answer as no solution or the empty set, ∅.
Can a solution be a fraction or a negative number?
Yes. Solutions are not always whole numbers. 3x = 2 has the solution x = 23, and x + 9 = 4 has the solution x = −5. If a word problem counts things, such as people or tickets, a fraction may mean the setup or the arithmetic needs another look.
What is a solution set?
The solution set is the collection of all solutions. For x2 = 9, it is {−3, 3}. For x > 7, it is every number greater than 7, often drawn on a number line. For the line y = 2x − 1, it is every point on the line.
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