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# Identities and equations with no solutions

If solving an equation yields a statement that is false, like 4 = 3, then the equation has no solution.

If solving an equation yields a statement that is true for a single value for the variable, like x = 3, then the equation has one solution.

If solving an equation yields a statement that is always true, like 3 = 3, then the equation has infinitely many solutions and is called an identity.

How many solutions does this equation have?

5c + 2c = 7c

Solution:

Solve for c.

 5c + 2c = 7c 7c = 7c Combine like terms 0 = 0 Subtract 7c from both sides

The statement 0 = 0 is true for every value of c. So, the equation is an identity and has infinitely many solutions.

How many solutions does this equation have?

-3x+3x = 0

no solution
one solution
infinitely many solutions