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Solving One-Step Linear Equations (page 2 of 4)

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


The "undo" of multiplication is division. If something is multiplied on the x, you undo it by dividing both sides (that is, dividing each term on both sides) of the equation by whatever is multiplied on the x:

  • Solve 2x = 5

    Since the x is multiplied by 2, I need to divide both sides by 2:

      2x/2 = 5/2, so x = 5/2

    Then the solution is x = 5/2 or x = 2.5.  

Warning: Usually the fractional form is the preferred form for your answers, rather than the decimal form; usually texts (and teachers) will prefer the "five-halves" answer over the "2.5" answer. If in doubt, check with your instructor.

The "undo" of division is multiplication:

  • Solve x/5 = –6

    Since the x is divided by 5, I'll want to multiply both sides by 5:

      solving animation: x = -30

    Then the solution is x = –30.

In the above solution (displayed in the animation), I multiplied by 5 on the right-hand side of the equation, and by 5/1 on the left-hand side. Since 5 = 5/1, this was a legitimate thing to do; I was being "fair" and doing the same thing to both sides of the equation. But why did I do it? Because it is often easier to keep track of what you're doing, when working with fractions, if all the numbers involved are in fractional form. Since I was needing to cancel a 1/5 on the left-hand side, it was useful  to multiply by 5 in the form 5/1. Most students find this habit to be helpful, so try to cultivate it now.


There is one very important point to make now: The solution to an equation is the value that makes the equation "true". This fact allows you to check your solutions. All you have to do is plug them back into the original equation, and make sure that you end up with a true statement.

The first equation above was x + 6 = –3, and our solution was x = –9. To verify this solution, plug it back in, and see if it works:

  • Check x = –9 as a solution for x + 6 = –3:
    •      x + 6  =   –3
      [–9] + 6 ?=? –3

        –9 + 6 ?=? –3

          6 – 9 ?=? –3

              –3  =   –3 Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

The last line above, –3 = –3, is a true statement, so the solution "checks", and the answer is verified as being correct. The other solutions above can be checked in the same way:

 

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  • Check x = –2 for x – 3 = –5:
    •        x – 3   =   –5
         [–2] – 3 ?=? –5

           –2 – 3 ?=? –5

      –2 + (–3) ?=? –5

                
      –5    =  –5

  • Check x = 2.5 for 2x = 5:
    •      2x   =   5
      2[2.5] ?=? 5

             
      5   =   5

  • Check x = –30 for x / 5 = –6:
    •        x / 5  =   –6
      [–30] / 5 ?=? –6

                –6  =   –6

So all of the solutions "check".


There is one "special case" related to the "undoing multiplication" case above: When x is multiplied by a fraction, you "undo" this multiplication by dividing both sides of the equation by that fraction. To divide by a fraction, you flip-n-multiply. To isolate a variable that is multiplied by a fraction, just multiply both sides of the equation by the flip ("reciprocal") of that fraction. For example:

  • Solve 3/5 x = 10

    Since x is multiplied by 3/5, I'll want to multiply both sides by 5/3, to cancel off the fraction on the x. Many students find it helpful to also turn the 10 into a fraction, by putting it over 1.

      (5/3)(3/5)x = (10/1)(5/3)

        x = 50/3

    Then the solution is x = 50/3.

Usually, you'll have to solve more complicated equations....


You can use the Mathway widget below to practice solving a linear equation by multiplying or dividing. Try the entered exercise, or type in your own exercise. Then click "Answer" to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Clicking on "View Steps" on the widget's answer screen will take you to the Mathway site, where you can register for a free seven-day trial of the software.)

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Cite this article as:

Stapel, Elizabeth. "Solving One-Step Linear Equations." Purplemath. Available from
    http://www.purplemath.com/modules/solvelin2.htm. Accessed
 

 

 

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