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Solving One-Step Linear Equations (page 1 of 3)

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:

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 2006-2008 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:

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:

Then the solution is x = –2.

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:

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:

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

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:

• 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.

Then the solution is x = ⁵⁰/₃.

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

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