# Lesson HOW MANY SOLUTIONS can there be in a linear equation?

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How many solutions can there be in a linear equation , where c, b, and c are constants?

The first answer that leaps to mind is one solution. Usually, especially in regular school math, that is the case. But not always.

### Case 1. One Solution

All linear equations ax+b=c where a is not equal to zero, have one solution:

### Case 2. No Solutions

A linear equation can have no solutions. Example: 0x + 1 = 2. Since 0x is always 0, and 0+1 = 1, we have an impossible equation 1=2. Any linear equation with no solution always has zero (0) as the coefficient before x.

### Case 3. Infinitely many solutions

A linear equation can have infinitely many solutions. Example: 0x+4=4. Since 0x is always 0 regardless of x, any value of x satisfies this equation.

### Conclusions

So, be careful with problems that reduce to a linear equation. Beware that they may have no solutions or an infinite number of solutions.

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