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The process of finding all solution(s) to an equation is called "solving the equation." Solving an equation is like solving a crime—you need to make like Sherlock Holmes and collect your clues, analyze them, and deduce their meanings. Fortunately, we're referring to the Sherlock Holmes from Sir Arthur Conan Doyle's stories, not the one in the feature films, so you probably won't need to deliver any uppercuts.
To solve an equation, we transform it into simpler equivalent equations until we can easily read off the value(s) of the variable that make the equation true. Usually we'd like to get the variable all by its lonesome on one side of the equation, with a number on the other side.
That's the ideal scenario, anyway, but sometimes it becomes tricky. Luckily, "tricky" is our middle name. Shmoop "Tricky" Aagaard. It's Norwegian.
An equation makes a claim that two quantities are equal. It's like saying, for example, that a dozen hockey players is twelve hockey players. Never mind what they're all doing on the ice at once. If we put each quantity on one side of a balance scale, the scale will balance.
Consider the equation x = 5. This equation claims that x is equal to 5. That means the "weight" of x is the same as the "weight" of 5, so let's get visual and draw a picture.
We can create equivalent equations to x = 5 by adding or subtracting the same amount of weight to or from each side of the scale. If we add a hacky sack with weight 1 to each pan, the scale must still balance. (Yes, those are hacky sacks. Just go with it.) It now represents the equation x + 1 = 6...although it's much more difficult to play hacky sack this way.
If we add another hacky sack with weight 1 to each pan, the scale must still balance, now representing the equation x + 2 = 7. Where are we getting all these things?
If we remove two hacky sacks from each side, we're back where we started, representing the equation x = 5. Plus, we get our hacky sacks back. We were starting to worry.
Now we know how to solve equations such as x + 2 = 7. We want to isolate x on one side of the equation, so we take 2 from each side and see the scale balance at x = 5. The solution to the equation is 5.
Although the hacky sack analogy breaks down a bit with negative numbers, unless you have one of those newfangled "negative space hacky sacks," the idea is the same. We can add or subtract any number we like, as long as we add or subtract it from both sides of the equation. Otherwise, we'd be unbalancing the scale. The goal is to keep the scale balanced while getting the variable all by itself on one side of the equation.
What's the solution to the equation x – 3 = -1?
All we've gotta do is add 3 to both sides of the equation, which keeps everything balanced:
x – 3 = -1
x – 3 + 3 = -1 + 3
x = 2
Boom, done. We got x by itself on one side, so we know x is 2.
What's the solution to x + 5 = -2?
Subtract 5 from each side of the equation to get that x all alone:
x + 5 = -2
x + 5 – 5 = -2 – 5
x = -7
One way to keep track of what we're doing is to write the operation we're performing under each side of the equation. Doctors will often do this sort of thing when they're performing operations:
One of the nice things about solving equations is that we can check our answers without sneaking into the principal's office to nab the solution key. When we solve an equation, we get a value that's supposed to be a solution to the equation. To check our answer, we stick that solution back into the equation and see if it makes the equation true. If not, you can rip off its mask and expose it for the fraud that it really is. "I would have gotten away with it, too, if it wasn't for you meddling kids!"
Here's how it works: we evaluate the left-hand side of the equation at the given value, evaluate the right-hand side of the equation at the given value, and see if the numbers match. We sometimes call this the "plug and chug" method. Plug in the value you found for the variable, and chug out the answer by simplifying the expressions on both sides of the equation.
You can also employ the "plug and chug and hug" method, which involves warmly embracing the correct answer. This one's not for you germaphobes out there.
Solve the equation 3 = y – 2. Check your answer.
To solve the equation, add 2 to both sides to find that 5 = y. To check the answer, we look to see if 5 really is a solution to the equation 3 = y – 2. That means it's plug and chug time: go back to the original equation and plug in y = 5.
3 = y – 2
3 = (5) – 2
3 = 3
The left-hand side of the equation is 3. The right-hand side evaluated at y = 5 is 5 – 2 = 3. Since the left-hand side and right-hand side agree with each other, y = 5 really is the solution to the equation. This is remarkably impressive, considering it's nearly impossible to get those two sides to agree on anything.
Just as we can add and subtract constants from both sides of an equation, we can also add and subtract copies of the variable from both sides of the equation. Therefore, if the same variable appears on both sides of the equation, we can reduce them as much as possible in order to get one variable all alone on one side. It's always nice to have just a single "x" (especially when following a treasure map, as you do).
Remember that our mission, if we choose to accept it, is to get the variable on one side of the = sign and a number on the other side.
Solve the equation 4x = 5x + 1. Check your answer.
We'd like to have all the x's by themselves on one side of the equation, so we subtract 4 copies of x from each side:
4x = 5x + 1
4x – 4x = 5x + 1 – 4x
0 = x + 1
Yay—so few copies! This will shave a bundle off our Kinko's bill.
We know what to do from here: subtract 1 from each side of the equation.
0 – 1 = x + 1 – 1
-1 = x
To check our answer, we evaluate the left side of the original equation and the right side of the original equation individually for x = -1. The left side of the equation evaluated at x = -1 is:
4(-1) = -4
The right side of the equation evaluated at x = -1 is:
5(-1) + 1 = -5 + 1 = -4
Because the two sides of the equation agree when x = -1, the solution to the equation is indeed x = -1. There's one of those negative solutions again. Sorry, Diophantus.
Solve the equation 8x = 5x + 24.
The first thing we do is subtract 5x from each side to find that 3x = 24.
We haven't talked yet about what to do with this sort of equation, but you can figure it out by thinking of the balance scale. If three copies of x weigh a total of 24, then each x must weigh one-third of 24. That is, x = 24 ÷ 3 = 8. Hopefully we aren't still talking about hacky sacks, because our ankles probably can't handle that much weight.
In other words, we're dividing both sides of 3x = 24 by 3 to get x by itself.
3x = 24
3x ÷ 3 = 24 ÷ 3
x = 8
Here's another way to think about it: to get x alone, we multiplied both sides of the equation by . It's no coincidence that this is the multiplicative inverse of 3. Did you have an "aha!" moment? Because we did.
In general, whenever we find ourselves with an equation of the form (coefficient) x = (some value), we multiply both sides of the equation by the reciprocal of the coefficient. Since any number times its reciprocal equals 1 (just try finding a number where this doesn't work—we double-dog dare you), this leaves us with an equation that has x all by itself on one side, and the solution to the equation on the other side.
As with addition and subtraction, multiplying or dividing both sides of an equation by the same quantity is okay, so long as we're not dividing by zero. Remember: it's all about doing to one side what you do to the other. If you give the left side a treat, you better have brought enough to share with the whole class.
Be careful: When solving an equation, whatever operation you perform on one side you must also perform on the other side. We know we've already said this, but we're hoping the 47th time's the charm. If you add 5 to the left-hand side of an equation, you must add 5 to the right-hand side of the equation. If you divide the right side of the equation by 3, you also need to divide the left side of the equation by 3.
What's the solution to 5x – 10 = 3x + 8?
First, get all the x's on one side of the equation by subtracting 3x from each side:
2x – 10 = 8
Then add 10 to both sides:
2x = 18
Finally, divide each side of the equation by 2:
x = 9
Recall that dividing by 2 is the same as multiplying by one-half. If you'd like to see someone give you the fish-eye, try ordering a footlong from Subway and then asking if they can multiply it by one-half.
We can think of dividing both sides of an equation by 2, or we can think of multiplying both sides of an equation by . As long as we do the arithmetic right, either way we'll find the same answer in the end. However, when fractions are involved, it's usually better to think of multiplying by a multiplicative inverse than dividing by a fraction.
Solve the equation
We want y all by itself, so multiply both sides of the equation by the multiplicative inverse of , which is . That'll cancel out the fraction on the left side of the equation:
y = 10
If you'd rather think of dividing both sides of the equation by , by all means, do so. Be on the lookout for any division snakes that might try to bite your ankles. Warning: these snakes may come disguised as numbers.
One thing that can trip us up if we aren't careful is notation with negative signs. When -x shows up in an equation, x is multiplied by -1. If it's helpful for you to re-write -x as (-1)x, do it. If anyone laughs at you for it, give us their name and the situation will be taken care of. We're not saying we'll do something to them, we're just saying...taken care of.