SAT Mc Graw Hill 2011

304 MCGRAW-HILL’S SAT

Systems

A system is simply a set of equations that are true at
the same time, such as these:

Although many values for xand y“satisfy” the first
equation, like (4, 0) and (2, −3) (plug them in and
check!), there is only one solution that works in both
equations: (6, 3). (Plug this into both equations and
check!)

The Law of Substitution

The law of substitutionsimply says that if

two things are equal, you can always substitute

one for the other.

Example:

The easiest way to solve this is to substitutethe sec-
ond equation (which is already “solved” for y) into the
first, so that you eliminateone of the unknowns. Since
y=x+1, you can replace the yin the first equation
with x+1 and get
3 x+(x+1)^2 = 7
FOIL the squared binomial: 3 x+x^2 + 2 x+ 1 = 7
Combine like terms: x^2 + 5 x+ 1 = 7
Subtract 7: x^2 + 5 x− 6 = 0
Factor the left side: (x+6)(x−1) = 0
Apply the Zero Product Property: x=−6 or x= 1
Plug values back into 2nd equation:
y=(−6) + 1 =− 5
or y=(1) + 1 = 2
Solutions: (−6, −5) and (1, 2) (Check!)

Combining Equations

If the two equations in the system are alike

enough, you can sometimes solve them more

easily by combining equations. The idea is sim-

ple: if you add or subtract the corresponding

sides of two true equations together, the result

should also be a true equation, because you are

adding equal things to both sides. This strategy

can be simpler than substitution.

37

1

xy^2

yx

+=

=+

⎧

⎨

⎩

3212

315

xy

xy

−=

+=

Example:

Add equations: 5 x = 30

Divide by 5: x= 6

Plug this back in and solve for y:

2(6) − 5 y= 7

Simplify: 12 − 5 y= 7

Subtract 12: − 5 y=− 5

Divide by 5: y= 1

Special Kinds of “Solving”

Sometimes a question gives you a system,

but rather than asking you to solve for each

unknown, it simply asks you to evaluate another

expression. Look carefully at what the question

asks you to evaluate, and see whether there is a

simple way of combining the equations (adding,

subtracting, multiplying,dividing) to find the

expression.

Example:

If 3x− 6 y=10 and 4x+ 2 y=2, what is the value of

7 x− 4 y?

Don’t solve for xand y! Just notice that 7x− 4 yequals

(3x− 6 y) +(4x+ 2 y)= 10 + 2 =12.

“Letter-Heavy” Systems

An equation with more than one unknown, or a

system with more unknowns than equations, is

“letter-heavy.” Simple equations and systems

usually have just one solution, but these “letter-

heavy” equations and systems usually have more

than one solution, and you can often easily find

solutions simply by “plugging in” values.

Example:

If 2m+ 5 n=10 and m≠0, then what is the value

of

You can “guess and check” a solution to the equation

pretty easily. Notice that m=−5, n=4 works. If you

plug these values into the expression you’re evaluating,

you’ll see it simplifies to 2.

4

10 5

m

− n

?

25 7

35 23

xy

xy

−=

+=

⎧

⎨

⎩

Lesson 2: Systems

Adding the correspond-

ing sides will eliminate

the y’s from the system