How To Know When An Equation Has NO Solution, Or Infinitely Many Solutions

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

Step 1

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.

Step 2

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).

Step 3

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.

Step 4

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.

Step 5

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!!!

Step 6

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.

Step 7

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.