248 MCGRAW-HILL’S SAT
Lesson 4:SimplifyingProblems
Beeline, Substitute, Combine, and Cancel
When a problem seems overwhelming, try one
of these four simplification strategies: beelining,
substituting,combining,and canceling.
Look for the Beeline—The Direct Route
Many SAT problems have “beelines”—direct
paths from the given information to the answer.
We sometimes miss the “beeline” because we
get trapped in a knee-jerk response—for in-
stance, automatically solving every equation
or using the Pythagorean theorem on every
right triangle. Avoid the knee-jerk response.
Instead, step back and look for the “beeline.”
If and what is the value of?
This problem looks tough because of all the un-
knowns. You might do the knee-jerk thing and try to
solve for a, b,and c. Whoa, there! Step back. The ques-
tion doesn’t ask for a, b,and c. It asks for a fraction that
you can get much more directly. Notice that just mul-
tiplying the two given fractionsgets you almost there:
. This is close to what you want—all
you have to do is divide by 3 to get. Substituting
the given values of the fractions gives you
which is the value of.
Simplify by Substituting
The simplest rule in algebra is also the most
powerful: Anything can be substituted for its
equal. When you notice a complicated expres-
sion on the SAT, just notice if it equals some-
thing simpler, and substitute!
If 3x^2 + 5x+ y = 8 and x≠0, then what is the value
of?
Again, take a deep breath. Both the equation and
the fraction look complicated, but you can simplify by
16 2
352
−
+
y
xx
a
10 c
1
4
33
1
4
×÷=,
a
10 c
3
25
3
10
a
b
b
c
a
c
×=
a
10 c
b
5 c
= 3 ,
3
2
1
4
a
b
=
just remembering that anything can be substituted for its
equal. Notice that 3x^2 + 5xappears in both the equation
and the fraction. What does it equal? Subtract yfrom
both sides of the equation to get 3x^2 + 5x= 8 −y. If
you substitute 8 −yfor 3x^2 + 5xin the fraction, you get
Nice!
Simplify by Combining or Canceling
Many algebraic expressions can be simplified
by combining or canceling terms. Always keep
your eye out for like termsthat can be com-
bined or canceled and for common factors in
fractionsthat can be canceled.
If mand nare positive integers such that m> nand
what is the value of m+ n?
To simplify this one, it helps to know a basic factor-
ing formula from Chapter 8, Lesson 5: m^2 – n^2 = (m – n)
(m+ n). If you factor the numerator and denominator
of the fraction, a common factor reveals itself, and it
can be canceled:
Since m+ nmust equal 9.
If f(x) = 2x^2 – 5x+ 3 and g(x) = 2x^2 + 5x+ 3, then for
how many values of xdoes f(x) = g(x)?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
Remember the simple rule that anything can be
substituted for its equal, and then cancel to simplify.
Since f(x) = g(x), you can say that
2 x^2 – 5x+ 3 = 2x^2 + 5x+ 3.
Subtract 2x^2 and 3: − 5 x= 5x
Add 5x:0 = 10x
Divide by 10: 0 = x
So the answer is (A) 0, right? Wrong! Remember that
the question asks for how many values of xare the
function values equal. Since we only got one solution
for x, the answer is (B) 1.
mn+
=
2
9
2
,
mn
mn
mnmn
mn
(^22) mn
22 2 2
−
−
=
−+
−
=
()()+
()
.
mn
mn
22
22
9
2
−
−
= ,
16 2
8
28
8
2
−
−
=
−
−
=
y
y
y
y