Example:
5x = 20
5x/5 = 20/5
x = 4
On the SAT, we can often use equations to answer a question even when we can’t solve the equation for each variable individually.
Example:
We might be told that (a + b)/10 = 15. How can we figure out the value of a + b? In school, you might try to figure out a first, and then b, and then add them together. But we don’t have enough information to do that. So what can we do? Well, we just solve for the entire amount a + b. In this situation, we can do that by multiplying both sides by 10, so a + b = 150. In this case, even though we can never know the individual values of a and b, we can know the sum a + b.
On the SAT, we can also solve equations “in terms of” one particular variable. To do this, we just isolate the target variable on one side of the equation.
Example:
What if we have to solve this expression in terms of n?
4n + 7y = 2a
4n = 2a – 7y
n = (2a – 7y)/4
n = (2a – 7y)/4
Sometimes you’ll have a “system” of equations. A system of equations contains two or more equations with the same variables.
Example:
This is a system of equations:
x + y = 5
2x – y = 7
The easiest way to solve a system of equations is to solve one equation in terms of one variable, like we just did before. Then we substitute in the second equation and solve.
Example:
First, we’ll isolate the y in the first equation, giving us that equation in terms of y: y = 5 – x. Now that we know y is the same thing as 5 – x, we just plug in 5 – x where y appears in the second equation:
2x – (5 – x) = 7
2x – 5 + x = 7
3x – 5 = 7
3x = 12
x = 4
Now that we know x is 4, we just plug that back into the first equation, and we’ll be able to solve for y:
4 + y = 5
y = 1
Inequalities
On the SAT, inequalities are statements that show a particular amount may be greater than or less than a second amount. They use these symbols:
The symbol < means “less than.”
The symbol > means “greater than.”
The symbol < means “less than or equal to.”
The symbol > means “greater than or equal to.”
You solve an inequality the same way you solve an equation, with one difference: when you multiply by -1 to solve for a variable, you have to switch the direction of the inequality symbol.
Example:
-x/4 = 10 -x/4 < 10
-x = 10(4) -x < 10(4)
-x = 40 -x < 40
x = -40 x > -40
Solving quadratic equations by factoring
A quadratic equation is an equation that involves three terms:
oone term is a variable expression raised to the power of 2.
oone term is a variable expression not raised to any power.
oone term is a regular number with no variable.
Example:
x2 + 3x = -2 is a quadratic equation because it involves a term with x squared, a term with x, and a regular number.
There is only one way to solve quadratic equations on the SAT, and that is by factoring. (See the discussion of factoring above).
To solve a quadratic equation by factoring, we have to make one side of the equation equal to zero, and then factor the other side of the equation (the quadratic part).
Example:
x2 +3x = -2
x2 +3x + 2 = 0
(x + 1)(x + 2) = 0
Now that we know (x + 1)(x + 2) = 0, what else do we know? We know that one of those two factors has to equal zero—either x + 1 = 0 or x + 2 = 0. How do we know this? Remember that the only way to multiply two numbers and get zero is if one of the numbers is zero. So if we can multiply x + 1 by x + 2 and get zero, then either x + 1 is zero or x + 2 is zero.
Once we’ve factored, we solve for the variable by creating two small sub-equations in which each factor is set equal to zero.
x + 1 = 0 or x + 2 = 0
x = -1 or x = -2
So in the equation x2 + 3x = -2 , x can equal either -1 or -2.
Quadratic equations can have multiple solutions, as we’ve just seen.
Functions
Functions are formulas that tell you how to generate one number by using another number.
Functions can be written in a lot of ways. On the SAT, they’ll usually be written in f(x) notation, also called “function notation.”
Example:
f(x) = x3 + 4 is a function written in function notation.
When we write with function notation, we don’t have to use f(x) specifically. We could write g(n), a(b), or whatever.
Don’t confuse function notation like f(x) with the multiplicative expression (f)(x), which means “f times x”!
When we evaluate a function for a certain number x, it means that we plug the number x into the function and see what the f(x) is.
Example:
If our function is f(x) = x3 + 4 and we want to evaluate the function where x = 2, then we get this:
f(x) = x3 + 4
f(2) = (2)3 + 4
f(2) = 8 + 4