"Linear" equations are equations with just a plain old variable like "x", rather than something more complicated like x^{2}, or ^{x}/_{y}, or square roots, or other more-complicated expressions. 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: □ + 3 = 5
Fill in the box: □ + 3 = 5
Once you'd learned your addition facts well enough, you knew that you had to put a "2" inside the box.
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Solving equations works in much the same way, but now we have to figure out what goes into the x, instead of what goes into the box. However, since we're older now than when we were filling in boxes, the equations can also be much more complicated, and therefore the methods we'll use to solve the equations will be a bit more advanced.
In general, to solve an equation for a given variable, we need to "undo" whatever has been done to the variable. We do this in order to get the variable by itself; in technical terms, we are "isolating" the variable. This results in the equation being rearranged to say "(variable) equals (some number)", where (some number) is the answer they're looking for. For instance:
The variable is the letter x. To solve this equation, I need to get the x by itself; that is, I need 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 this 6 to get rid of it. That is, I will need to subtract a 6 from the x in order to "undo" their having added a 6 to it.
This brings up the most important consideration with equations:
No matter what kind of equation we're dealing with — linear or otherwise — whatever we do to the one side of the equation, we must do the exact same thing to the other side of the equation. Equations are like toddlers in this respect:
We have to be totally, totally fair to the two sides, or unhappiness will ensue!
Whatever you do to an equation, do the EXACT SAME thing to BOTH sides of that equation!
Probably the best way to keep track of this subtraction of the 6 from both sides is to format your work this way:
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What you see here is that I've subtracted 6 from both sides, drawn a horizontal "equals" bar underneath the entire equation, and then added down. On the left-hand side (LHS) of the equation, this gives me:
x plus nothing is x, and 6 minus 6 is zero
On the right-hand side (RHS) of the equation, I have:
–3 plus –6 is –9
The solution is the last line of my work; namely:
x = –9
The same "undo" procedure works for equations in which the variable has been paired with a subtraction.
The variable is on the left-hand side (LHS) of the equation, and it's paired with a "subtract three". Since I want to get x by itself, I don't like the "3" that's currently subtracted from it. The opposite of subtraction is addition, so I'll undo the "subtract 3" by adding 3 to both sides of the equation, and then adding down to simplify to get my answer:
Then my answer is:
x = –2
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You may be instructed to "check your solutions", at least in the early stages of learning how to solve equations. To do this "checking", you need only plug your answer into the original equation, and make sure that you end up with a true statement. (This is, after all, the definition of the solution to an equation; namely, the solution is any value, or set of values [for more complicated equations, later on], which makes the original equation a true statement.)
So, to check my solution to the above equation, you'd plug "–2" in place of the x in left-hand side (LHS) of the original equation, and check that this simplifies to give the original value for the right-hand side (RHS) of the equation:
Checking:
LHS: (–2) – 3 = –5
RHS: –5
Because each side of the original equation now evaluates to the exact same thing, this confirms that the solution is indeed correct.
This time, the variable is on the right-hand side (RHS) of the equation. That's okay; it doesn't matter where the variable is, as long as I get isolate it (that is, as long as I can get it by itself on one side of the "equals" sign).
In this equation, I've got a three that's subtracted from the variable. To undo the subtraction, I'll add three to either side of the equation.
4 = x - 3
+3 + 3
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7 = x
(I could have written the right-hand side, after adding down, as "x + 0", but "plus zero" is customarily ignored. That's why I carried down only the x on the right-hand side.)
Now, as part of my hand-in work, I need to show that I've checked this solution by plugging it back into the RHS of the original equation, and confirming that I end up with the LHS of the original equation; that is, that I end up with 4:
Checking:
RHS: (7) – 3 = 4 = LHS
The "checking" part is what I just did above. I've made sure to label things clearly, so the grader is able to find my "check" (so I'll get full credit on the exercise). My final answer is:
x = 7
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When I solved the last exercise above, the variable had ended up on the right-hand side of the "equals" sign. But in my solution, I wrote the answer with the variable on the left-hand side of the "equals" sign. This is pretty standard. When you're solving, the variable will end up wherever it ends up. When you're writing out the solution, the variable goes on the left. Why? Because.
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This equation is almost solved. But not quite. I don't have plain old x on the right-hand side; instead, I've got –x. What to do?
I can kind-of think of the –x as being 0 – x. So what would happen if I added x to each side of the equation?
2 = –x
+x +x
-------
x + 2 = 0
Okay; that helped. By taking the variable and "adding it over to the other side", I've now got the variable in a format I like. And this has also converted the original equation into a simple one-step equation. I'll get rid of the 2 from the left-hand side by "subtracting it over to" the right-hand side:
x + 2 = 0
-2 = -2
----------
x = -2
This answer makes sense. If the negative of the variable equalled a positive two, then the positive of the variable should equal a negative two. So my answer is:
x = –2
Technically, that last example was a two-step equation, because solving it required adding one thing to both sides of the equation, and then subtracting another thing to both sides. The important thing to notice is that you can add and subtract variables to the other side of an equation, just like you can add and subtract numbers to the other side. The exact same methods work with both variables and numbers.
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 button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)
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