Lesson 3-5: Determine the Number of Solutions

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.

Property

A linear equation in one variable can have one of three possible outcomes for the number of solutions:

• Exactly One Solution: The equation simplifies to $x = a$, where $a$ is a specific number.
• No Solution: The equation simplifies to a false statement, such as $a = b$ (e.g., $3 = 5$).
• Infinitely Many Solutions: The equation simplifies to a true statement, such as $a = a$ (e.g., $4 = 4$).

Examples

Property

For a simplified linear equation in the form $ax + b = cx + d$:

• If $a \neq c$, the equation has exactly one solution.
• If $a = c$ and $b = d$, the equation has infinitely many solutions.
• If $a = c$ and $b \neq d$, the equation has no solution.

Examples

Property

An identity is an equation that is always true. It has infinitely many solutions.

Examples

$10 - 6x = -2(3x - 5)$ simplifies to $10 - 6x = -6x + 10$, which becomes $10 = 10$. It's an identity!
$2(x + 3) = 2x + 6$ simplifies to $2x + 6 = 2x + 6$. This is always true, so it's an identity.

Explanation

An identity is a math statement that’s always agreeable, no matter what number you plug in for the variable. When you simplify it, the variable vanishes and you're left with a true statement like $10 = 10$. This means every number is a solution! It’s universally true, like pizza being delicious.