Solving Simple and Intermediate Algebra Equations
Solving equations is the main event in algebra. In this section, you use the tools given in Chapter 3 to solve a variety of increasingly complex algebraic equations.
Solving basic algebraic equations
You can solve some algebraic equations just by inspection — that is, just by looking at them and thinking about what the variable must be. For example, the following four algebraic equations are progressively more difficult. How many can you solve without writing anything down?
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If you struggled and finally gave up on at least one of these equations, you can probably see the need for a more systematic method for solving equations that arise when building 60-story towers, mapping the human genome, or sending rockets to Saturn.
You start by solving basic equations by isolating the variable — working step by step to discover the value of the variable by getting it alone on one side of the equals sign. When the variable is left standing by itself with no place to hide, whatever’s left on the other side of the equals sign is the solution.
Each step along the way, you use the balance scale which encourages you to think of an equation as a scale with two perfectly balanced sides: If you remove the same amount of weight from both sides of the scale, it stays in balance. In an equation, when you perform the same operation on both sides, they both remain equal. Here is an example:
If 
, then what does 
equal?
(A) 10
(B) 11
(C) 12
(D) 13
This is a common type of SAT question that asks you to solve one equation, and then plug that answer into an expression and evaluate the result. Begin by solving the equation. To make it simpler, add to both sides:
At this point, you may already see the answer, but if you have any doubt, divide both sides by 12:
You’re not done! The question isn’t asking for the value of but rather the value of 
:
Therefore, Answer D is correct.
As another example, consider the following question, which includes an equation from earlier in this section:
What is the value of x in this equation?
Again, you want to solve the equation and then plug the result into an expression. This time, begin by adding to both sides:
Now subtract 26 from both sides:
To finish up, divide by 14:
Therefore, the correct answer is –3.
More complicated equations may require you to distribute in order to remove parentheses on one or both sides of the equation before you can combine like terms and solve. Here is an example:
What is the value of x in this equation?
To solve this equation, begin by dropping the parentheses on the left side of the equation and distributing on the right side:
Next, simplify both sides of the equation:
Now, solve the equation:
Therefore, the correct answer is 8.
Working with equations that have more than one variable
In most cases, you can’t solve an equation that has more than one variable, because it produces infinitely many solutions. But SAT questions like these skirt around this issue in a variety of ways. In this section, I give you a few practical pointers for answering these types of questions.
Isolating a specific variable
One common type of SAT math question presents you with an equation in two or more variables and asks you to solve it for a given variable. To do this, isolate that variable on one side of the equation and get all of the other variables over to the other side. Here is an example:
If 
, what is the value of a in terms of and
(A) 
(B) 
(C) 
(D) 
To solve, isolate the variable a. Begin by multiplying both sides of the equation 
by
Next, subtract 
from both sides:
To finish, divide both sides by 2:
Therefore, Answer D is correct.
A common type of SAT math question gives you a geometric formula and asks you to solve it for a specific variable. Here is an example:
This formula is for the area A of a trapezoid based on its height h and the lengths of its two bases, 
and 
. Which of the following allows you to find the length of base 
based on the values of the other three variables?
(A) 
(B) 
(C) 
(D) 
To solve for 
, begin by dividing both sides of the equation by
Next, multiply both sides by 2:
Finally, subtract 
from both sides:
Therefore, Answer D is correct.
Factoring to solve for a variable
In some cases, when solving an equation for a variable, you may need to factor to isolate that variable. Here is an example:
If 
, what is the value of y in terms of
(A) 
(B) 
(C) 
(D) 
To begin, isolate the terms that include y on one side of the equals sign:
Now, factor out y on the left side of the equation:
To finish up, divide out 
on both sides. I do this in two steps, so you can see how it works:
Therefore, Answer B is correct.
Solving for an expression that contains more than one variable
In some cases, an SAT question will give you an equation that has more than one variable and ask you to find the value of an expression that includes all of the variables. In these cases, you need to think of the two variables as a single unit and figure out how to produce the expression that the question provides, as in this example:
If 
, what is the value of 
?
Your obvious first step is to subtract 5 from both sides:
Now, you may feel stuck, because you can’t solve for either a or However, step back from the equation and notice that the question is asking you for the value of 
. Thus, you can substitute 46 for 
and evaluate:
Therefore, the correct answer is 89.
Here’s another example:
If 
, what does 
equal?
This time, begin by subtracting 17 from both sides:
As with the previous example, you can’t solve this equation for either variable. The insight here is to see that you can build 
by dividing each side of the equation by 2:
To complete the problem, substitute 6 for 
into the expression 
:
Therefore, the correct answer is 16.
Keep an eye open for opportunities to use these tricks to answer SAT math questions!
