Description:
Solve for x questions require the use of basic algebra to solve for variables.
The harder questions involve two variables, which may require either substitution
or linear addition to solve.
Approach:
Solve for one variable at a time. If the question has two equations, stack them
and see if adding them will eliminate the variable you don't need.
1) Basic Algebra Strategy
-
Isolate the variable
-
To separate the variable from the numbers around it, perform the opposite
of every process in the equation.
-
Solve for the variable
-
To check your work, substitute the answer for the variable and make
sure the problem works out.
2) Practice:
-
3x + 5 = 26
-
16 - 4r = 64
-
2q + 6 = 30 - q
3) FOIL (First Outside Inside Last)
Remember this tried and true acronym? It helps you multiply two algebraic factors.
For example:
-
(x - 6)(x + 5)
-
(x * x)
+ (x * 5) + (-6 * x) + (-6 * 5)
-
x2
- x - 30
4) FOIL the expressions:
-
(x + 8)(x + 5) =
-
(x - y)(x + y) =
5) Multiple Variables
SAT problems with two equations and two variables can usually be solved by
stacking the equations and eliminating the variable you don't need.
For example:
-
2x + 4y = 10
-
6x + 2y = 15
If you need to solve for x, multiply one of the equations by a constant so
that the y
terms drop out when you add the equations. In this case we should multiply
the second equation by negative 2, and then add the two equations.
-
2x + 4y = 10
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-12x -4y = -30
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-10x = -20
-
x = 2
6) Substitution
Substitution is another option for solving multiple equations with more
than one variable. Solve for one of the variables in terms of the other,
and then plug this value back into one of the original equations to solve
for the other variable.
For example:
-
Solve for y
using the first equation.
-
y = 14 -2x
-
Plug this value into the second equation to solve for x.
-
3x + 3(14 - 2x) = 27
-
3x + 42 - 6x = 27
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-3x + 42 = 27
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-3x = -15
-
x = 5
-
Plug the value of x
into the first equation to solve for y.
-
2x + y = 14
-
x = 5
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2(5) + y = 14
-
10 + y = 14
-
y = 4
7) Solving for Combinations of Variables
Many multiple variable questions on the
SAT ask you to solve for a
combination of variables rather than for each individual variable.
For example:
If x = 12 and
x + y = 3;
what is the value of
x2 + xy?
We are used to solving for each variable individually, and then using
them together to solve for the entire expression. However, very
often on the SAT you have to
combine the individual expressions to get the answer.
-
Add the expressions together.
-
Multiply the expressions together.
-
Divide the expressions by one another.
-
Subtract one expression from the other.
For example:
In the above problem we just need to multiply the two
expressions together to get the answer we need.
-
x * (x + y) =
x2 + xy
-
x = 12,
and x + y = 3
-
Therefore, x2 + xy = 12 * 3 =
36
Combining the expressions in this way is often much simpler and
quicker than solving for each variable individually.
8) Quadratic Equations
The basic format of a quadratic equation is:
ƒ(x)
= ax2+ bx + c
Quadratics can usually be factored and broken down into two expressions
that identify the solutions of the equation.
Factor and solve:
x2
+ 5x + 6 = 0
What are the two possible values of x?
You can also solve a quadratic like the one above by graphing the equation
on your calculator and finding the x intercepts.
9) Common Quadratic Formulas must be Memorized
| Factors |
Quadratic formula |
|
(x + y)(x + y) aka (x + y)2
|
x2 + 2xy + y2
|
| (x - y)(x - y) aka (x - y)2 |
x2 - 2xy + y2 |
| (x + y)(x - y) |
x2 - y2 |
10) Practice:
-
If (x + y) = 20
and (x2 - y2) = 100,
what is the value of (x - y)?
-
If 3x2 + y = 10
and xy = 2,
what is the value of
3x2y + xy2?
