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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 problems like the following:

Fill in the box:

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. For instance: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

• Solve x + 6 = –3

I want to get x by itself; that is, I want to get "x =" on one side, and whatever numbers 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 with 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, in order to "undo" having added a 6.

This brings up an important point: 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 little children: You have to be totally, totally fair! 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. Then the solution is 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 x, you undo it by dividing both sides (that is, dividing each term on both sides) 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.  

Usually the fractional form is the preferred form for your answers, rather than the decimal form. That is, usually texts (and teachers) will prefer the " ⁵/₂" answer, rather than 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.

There is one sort of "special case": When x is multiplied by a fraction, you can divide both sides by that fraction (since division is the "undo" of multiplication), but remember that, to divide by a fraction, you flip-n-multiply. So it's simplest to just multiply both sides of the equation by the flip ("reciprocal") of the 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. That is, 10 = ¹⁰/₁:

Then the solution is x = ⁵⁰/₃.

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

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