Solution of an equation
Identify the solution of an equation as the value that makes both sides equal: verify solutions by substitution and recognize that an equation may have zero, one, or infinitely many solutions.
Key Concepts
A number is a solution of an equation in one variable if substituting the number for the variable results in a true statement.
Is $x = 5$ a solution to $3x 2 = 13$? Check: $3(5) 2 = 15 2 = 13$. Yes, it's a true statement!|Is $y = 2$ a solution to $4y + 10 = 3$? Check: $4( 2) + 10 = 8 + 10 = 2$. No, because $2 \neq 3$.|To confirm the solution to $6n+1=13$ is $n=2$, substitute it back: $6(2)+1 = 12+1 = 13$. It checks out!
Finding a solution is like discovering the secret code that opens a lock! You are hunting for that one specific number that you can substitute for the variable to make both sides of the equation perfectly equal. If they match after you plug it in, you have successfully found the code. If they do not, the mystery remains unsolved.
Common Questions
What is the solution of an equation?
The solution of an equation is the value (or values) of the variable that make the equation a true statement when substituted. For 3x-5=7, substituting x=4 gives 3(4)-5=7, which is true, so x=4 is the solution.
How do you verify that a value is a solution to an equation?
Substitute the value into both sides of the original equation and simplify each side independently. If both sides yield the same number, the value is a valid solution. If the sides differ, the value is not a solution.
How many solutions can a linear equation in one variable have?
A linear equation in one variable has exactly one solution, no solution, or infinitely many solutions. If simplifying gives a false statement (0=5), there is no solution. If it gives a true statement (0=0), every real number is a solution (infinitely many). Otherwise there is exactly one solution.