Solving Equations
Solving Equations
To solve an equation, you have to isolate the variable you’re solving for. You have to “manipulate” the equation until you get the variable alone on one side of the equal sign. By definition, the variable is then equal to everything on the other side of the equal sign. You’ve just solved for the variable. Congratulations.
The Fine Art of Manipulation
You can’t manipulate an equation the way you manipulate your little brother or sister. When manipulating equations, there are rules. Here’s the first and most fundamental (it’s so important we’re going to bold it): Whatever you do to one side of an equation, you must do to the other side. If you divide one side of an equation by 3, divide the other side by 3. If you take the square root of one side of an equation, take the square root of the other. If you fall in love with one side of the equation, fall in love with the other. Neither side will think you’re a two-timer. They’ll think you’re a highly skilled mathematician.
By treating the two sides of the equation in the same way, you don’t change what the equation means. You change the form of the equation—that’s the point of manipulating it—but the equation remains true since both sides stay equal.
Take, for instance, the equation 3x + 2 = 5. You can do anything you want to it, anything at all, and as long as you do that thing to both sides of the equation, x will always equal 1. For example, if you subtract 2 from both sides,
3x + 2 – 2 = 5 – 2
3x = 3
x = 1
And if you multiply both sides by 2,
2(3x + 2) = 2(5)
6x + 4 = 10
6x = 6
x = 1
In addition to the “do the same things to both sides of the equation” rule that you must follow, there are other rules of manipulation that you should follow. Nothing will go horribly wrong if you don’t follow them, but it will take you longer to solve the question. Here are the rules:
  1. Combine like terms to make the equation simpler.
  2. Manipulate the equation in the reverse order of operations.
The second rule means that you should first subtract or add any extra terms on the same side as the variable. Then divide and multiply anything on the same side as the variable. Next, raise both sides of the equation to a power or take their roots according to any exponent attached to the variable. Finally, work out anything inside parentheses. Do the order of operations backward: SADMEP! The idea is to “undo” everything that has been done to the variable so that it will be isolated in the end. Example time:
In this equation, poor little x is being square rooted, multiplied by 2, added to 3, and encased in parentheses. You’ve got to get him out of there! Undo all of these operations in order to liberate x.
First, subtract 2 from both sides of the equation:
Then multiply both sides by 2 to get rid of the fraction:
Now divide both sides by 3 (later, parentheses):
Subtract 3 from each side:
Divide both sides by 2:
And, finally, square each side to get rid of the square root:
Success! You’ve freed x from all of those bullying operations.
Location. Location. Location.
Isolating for x is all about where x is located. A variable in the numerator of a fraction is actually a pretty easy location to isolate. But if x is in the denominator of a fraction, things get more complicated.
Following SADMEP, start by subtracting the 3:
But now you have to get the x out of the denominator, and the only way to do that is to multiply both sides of the equation by that denominator, x + 2:
Divide both sides by 4:
Subtract 2 from each side:
Simplification Tools
By now you know the rule: When solving an equation, never do something to one side of an equation that you don’t do to the other. If you add 4 to one side, you have to add 4 to the other. But what if there were some simplification tools that didn’t change the value of an expression? What if you could simplify one side of an equation without changing its value? That would rock. Why? Because it would allow you to make solving equations much simpler and save you time on the SAT.
The first step to adding ferocious simplification tools to your arsenal is the rule of distribution, which states
a can be any kind of term, meaning it could be a variable, a constant, or a combination of the two.
When you distribute a factor into an expression within parentheses, multiply each term inside the parentheses by the factor outside the parentheses. For example, in the previous problem, when you had 1 = 4(x + 2), you didn’t actually have to divide both sides by 4. You could have distributed the 4 and pushed off all those messy fractions until the end: 1 = 4x + 8; 4x = –7; x = –7/4. So, if you have the expression 3y(y2 – 6):
Seems logical enough. But the true value of distributing becomes clear when you see a distributable expression in an equation: 3y(y2 + 6) = 3y3 + 36 looks like it’d be hard to solve, since there aren’t any equal terms to add or subtract away. But wait a sec . . . what if you distribute that 3y?
Shocking revelation! It’s suddenly clear that you can subtract 3y3 from both sides:
Factoring an expression is the opposite of distributing. 4x3 – 8x2 is one mean-looking expression. Or so it seems, until you realize that all the terms share the greatest common factor 4x2, which you can factor out:
With distributing and factoring, you can group or ungroup quantities in an equation to make your calculations simpler. Here are a few more examples:
Combining Like Terms
After factoring and distributing, you can take additional steps to simplify expressions or equations. Combining like terms is one of the simplest techniques you can use. It involves adding or subtracting the coefficients of variables that are raised to the same power. For example, by combining like terms, the expression
can be simplified by adding the coefficients of the variable x3 (–1 and 3) together and the coefficients of x2 (1 and 4) together:
Variables that have different exponential values are not like terms and can’t be combined. Two terms that do not share a variable are also not like terms and cannot be combined regardless of their exponential value.
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