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Solving Linear Equations with Parentheses;
    Solving "No Solution" Equations and 
         "All Real Numbers" Equations 
(page 4 of 4)

Sections: One-step equations, Multi-step equations, "No solution" and "all x" equations


  • Solve 11 + 3x 7 = 6x + 5 3x

    First, combine like terms; then solve:

      '11 + 3x – 7 = 6x + 5 – 3x' yields '4 = 5'

    Then the "solution" is "no solution".

When you try to solve an equation, you are starting from the (unstated) assumption that there actually is a solution. When you end up with nonsense (like the nonsensical equation "4 = 5" above), this says that your initial assumption (that there was a solution) was wrong; in fact, there is no solution. Since the statement "4 = 5" is utterly false, and since there is no value of x that ever could make it true, then this equation has no solution.

Advisory: This answer is entirely unlike the answer to the previous exercise, where there was a value of x that would work. Don't confuse these two very different situations: "the solution exists and has the value of zero" is not in any manner the same as "no solution value exists at all".

And don't confuse the "no solution" type of equation above with the following type:

  • Solve 6x + 5  2x = 4 + 4x + 1
  • First, I'll combine like terms; then I'll solve:

      5 + 4x = 5 + 4x, subtract 4x from either side to get 5 = 5

    Is there any value of x that would make the above statement false? Isn't 5 always going to equal 5? In fact, since there is no "x" in the solution, the value of x is irrelevant: x can be anything I want. So the solution is "all x".

This solution could also be stated as "all real numbers" or "all reals" or "the whole number line"; expect some variation in lingo from one text to the next. Note that, if I had solved the equation by subtracting a 5 from either side of 5 + 4x = 5 + 4x to get "4x = 4x", I would have ended up with nothing other than another trivially-true statement. I could also have subtracted both 4x and 5 from both sides to get "0 = 0", but the solution would still be the same: "all x". Don't be surprised if, for "all real numbers" or "no solution" equations, you don't necessarily have the exact same steps as some of your fellow students. Since there are infinitely-many always-true equations (like "0 = 0") and infinitely-many nonsensical equations (like "3 = 4"), there will be many ways of arriving at these answers.


 

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  • Solve 9 = 3(5x 2)
  • Solve 6x (3x + 8) = 16

    Be careful with taking negatives through parentheses. If it helps you to put a "1" in front of the parentheses, then do so. Copyright Elizabeth Stapel 2002-2011 All Rights Reserved

    I"ll simplify on the left-hand side first; then I'll solve in the usual way:

      x = 8

    Then the solution is  x = 8.

  • Solve 7(5x 2) = 6(6x 1)

    I have to be sure to take the 7 and the 6 all the way through their respective parentheses.

      -8 = x

    Then the solution is  x = 8.

For this type of problem, take your time and write out all of your steps. Don't try to do everything in your head.

  • Solve 13 (2x + 2) = 2(x + 2) + 3x

    Multiply through the parentheses (a minus sign on the left, and a two on the right), combine like terms, simplify, and solve:

      1 = x

    Then the solution is  x = 1.


Don't forget: There is never any reason to be unsure of your solution: you can always check your answer to any equation-solving exercise! The point of a solution is that it is the x-value that makes the equation true. To check your answer, plug your solution back into the original equation, and make sure that the equation "works". For instance, in the last exercise above, my solution was x = 1. Here's the check:

      13  (2x + 2)   =   2(x + 2) + 3x
    13
     (2[1] + 2) ?=? 2([1] + 2) + 3[1]
    13
      (2 + 2)   ?=?   2(1 + 2)  +   3
    13      (4)      ?=?     2(3)      +   3

    13       4        ?=?      6         +   3

                  9         =        9

So the solution "checks", and I know that my answer is correct.

Advisory: This ability to check your answers can come in handy on tests. Once you've completed all the questions, go back and plug in your solutions. If the solution "checks", then you know you got that question right. If it doesn't check, then you have the chance to correct your mistake before you hand in the test!

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Cite this article as:

Stapel, Elizabeth. "Solving Linear Equations with Parentheses; Solving 'No Solution' Equations
    and 'All Real Numbers' Equations."  Purplemath. Available from
    
http://www.purplemath.com/modules/solvelin4.htm. Accessed
 

 



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