SAT Mc Graw Hill 2011

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

()

.