How to Know when an Equation has NO Solution, or Infinitely many Solutions

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Many students assume that all equations have solutions. This article will use three examples to show that assumption is incorrect.

    Given the equation 5x - 2 + 3x = 3(x+4)-1 to solve, we will collect our like terms on the left hand side of the equal sign and distribute the 3 on the right hand side of the equal sign.

    5x - 2 + 3x = 3(x+4)-1 is equivalent to 8x - 2 = 3x + 12 - 1, that is, 8x - 2 = 3x + 11. We will now collect all our x-terms on one side of the equal sign (it does not matter whether the x-terms are placed on the left side of the equal sign or on the right side of the equal sign).

    So 8x - 2 = 3x + 11 can be written as 8x - 3x = 11 + 2, that is, we subtracted 3x from both sides of the equal sign and added 2 to both sides of the equal sign, the resulting equation now is 5x = 13. We isolate the x by dividing both sides by 5 and our answer will be x = 13/5. This equation happens to have a unique answer, which is x = 13/5.

    Let us solve the equation 5x - 2 + 3x = 3(x+4) + 5x - 14. In solving this equation, we follow the same process as in steps 1 through 3 and we have the equivalent equation 8x - 2 = 8x - 2. Here, we collect our x-terms on the left side of the equal sign and our constant terms on the right side, thus giving us the equation 0x = 0 which is equal to 0=0, which is a true statement.

    If we look carefully at the equation, 8x - 2 = 8x - 2, we will see that for any x you substitute on both sides of the equation the results will be the same so the solution to this equation is x is real, that is, any number x will satisfy this equation. TRY IT!!!

    Now, let us solve the equation 5x - 2 + 3x = 3(x+4) + 5x - 10 following the same procedure as in the steps above. We will get the equation 8x - 2 = 8x + 2. We collect our x-terms on the left hand side of the equal sign and the constant terms on the right hand side of the equal sign and we will see that 0x = 4, that is, 0 = 4, not a true statement.

    If 0 = 4, then I could go to any bank, give them $0 and get back $4. No way. This will never happen. In this case, there is no x that will satisfy the equation given in Step #6. So the solution to this equation is: there is NO SOLUTION.

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