Number of Solutions to Linear Equations

When you solve a linear equation, you are usually looking for the value of the variable (like $x$) that makes the equation true. However, not every equation has exactly one answer. By simplifying an equation, you can determine whether it has one solution, no solution, or infinitely many solutions.

Here is how to tell them apart.

Exactly One Solution

Most linear equations you encounter will have exactly one solution. This happens when you can isolate the variable and it equals a specific number.

Example: How many solutions does $x + 4 = 2x - 1$ have?

Let's solve it:

1. Subtract $x$ from both sides: $4 = x - 1$
2. Add $1$ to both sides: $5 = x$

Because we ended up with a single, specific value for $x$ (which is $x = 5$), this equation has exactly one solution.

No Solution

Sometimes, an equation represents an impossible situation. If you simplify the equation and the variables cancel out completely, leaving you with a false statement (like $0 = 5$ or $3 = 5$), the equation has no solution. There is no number in the world you could plug in for $x$ to make it true.

Example: How many solutions does $2x + 3 = 2x + 5$ have?

Let's solve it:

1. Subtract $2x$ from both sides to group the variables: $3 = 5$

Wait, $3$ does not equal $5$! Because we are left with a false statement, this equation has no solution.

Infinitely Many Solutions

If you simplify an equation and the variables cancel out completely, but you are left with a true statement (like $0 = 0$ or $-3 = -3$), the equation has infinitely many solutions. This means that any real number you plug in for $x$ will make the equation true. The left side and the right side are actually the exact same expression.

Example: How many solutions does $3(x - 1) = 3x - 3$ have?

Let's solve it:

1. Distribute the $3$ on the left side: $3x - 3 = 3x - 3$
2. Subtract $3x$ from both sides: $-3 = -3$

Because $-3 = -3$ is always true, this equation has infinitely many solutions.

Summary

When solving a linear equation, simplify both sides and isolate the variable:

• One Solution: You get $x = \text{number}$ (e.g., $x = 5$).
• No Solution: You get a false statement (e.g., $3 = 5$).
• Infinitely Many Solutions: You get a true statement (e.g., $0 = 0$ or $-3 = -3$).