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Solving One-Step Linear Equations (page 1 of 4)

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

"Linear" equations are equations with just a plain old variable like "x", rather than something more complicated like x2 or x/y or square roots or such. Linear equations are the simplest equations that you'll deal with. You've probably already solved linear equations; you just didn't know it. Back in your early years, when you were learning addition, your teacher probably gave you worksheets to complete that had exercises like the following:

Fill in the box: 'box' + 3 = 5

Once you'd learned your addition facts well enough, you knew that you had to put a "2" in the box. Solving equations works in much the same way, but now you have to figure out what goes into the x, instead of what goes into the box. However, since you're older now, the equations can be much more complicated, and therefore the methods you'll use to solve the equations will be a bit more advanced.

In general, to solve an equation for a given variable, you need to "undo" whatever has been done to the variable. You do this in order to get the variable by itself; in technical terms, you are "isolating" the variable. This results in "(variable) equals (some number)", where (some number) is the answer they're looking for. For instance: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

  • Solve x + 6 = –3

    I want to get the x by itself; that is, I want to get "x" on one side of the "equals" sign, and some number on the other side. Since I want just x on the one side, this means that I don't like the "plus six" that's currently on the same side as the x. Since the 6 is added to the x, I need to subtract to get rid of it. That is, I will need to subtract a 6 from the x in order to "undo" having added a 6 to it.

    This brings up the most important consideration with equations: No matter what kind of equation you're dealing with -- linear or otherwise -- whatever you do to the one side, you must do the exact same thing to the other side! Equations are like toddlers in this respect: You have to be totally, totally fair!

    Whatever you do to an equation,
    do the   S A M E   thing
    to   B O T H   sides of that equation!

    Probably the best way to keep track of this subtraction of 6 from both sides is to format your work this way:

      animation: solving x + 6 = -3 to get x = -9

    What you see here is that I've subtracted 6 from both sides, drawn an "equals" bar underneath both sides, and added down: x plus nothing is x, 6 minus 6 is zero, and –3 plus –6 is –9. The solution is the last line of my work: x = –9.

The same "undo" procedure works for subtraction:

  • Solve x – 3 = –5

    Since I want to get x by itself, I don't want the "–3" that's with the variable. The opposite of subtraction is addition, so I'll undo the –3 by adding 3 to both sides, and then adding down:

      x - 3 + 3 = -5 + 3, so x = -2

    Then the solution is x = –2.

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

(Clicking on "Tap to view steps" on the widget's answer screen will take you to the Mathway site for a paid upgrade.)

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Cite this article as:

Stapel, Elizabeth. "Solving Linear Equations: 'One-Step' Equations." Purplemath. Available from Accessed



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