SAT Mc Graw Hill 2011

CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 307

Answer Key 2: Systems

7. (8.5, 3.5) a−b= 5
   +a+b= 12
   Add the equations: 2 a= 17
   Divide by 2: a=8.5
   Plug in to find b: (8.5) −b= 5
   Subtract 8.5: −b=−3.5
   Multiply by −1: b=3.5


8. (10/3, −6) − 3 x− 5 y= 20
   −(− 3 x− 4 y=14)
   Subtract the equations: −y= 6
   Multiply by −1: y=− 6
   Plug in to find x: − 3 x−5(−6) = 20
   Simplify: − 3 x+ 30 = 20
   Subtract 30: − 3 x=− 10
   Divide by −3: x=10/3


9. (−12, 15) Add the equations to get

Combine fractions:

Multiply by 12: 7 x=− 84

Divide by 7: x=− 12

Plug in and solve for y: y= 15

10. There are many solutions. Here are a few:
    (0, 8); (20, 0); (10, 4); (5, 6)


11. There are many solutions. Here are a few:
    (1, 6, 0); (3, 4, −4); (2, 5, 31); (7, 0, 24)

7

12

7

x

=−

xx

34

+=− 7

Concept Review 2

1. Any set of equations that are true at the same time.


2. Substitution and combination.


3. Plug the solutions back into the equations and
   check that both equations are true.


4. (−10, −8) Substitute: 3(y−2) − 4 y= 2
   Distribute: 3 y− 6 − 4 y= 2
   Combine: −y− 6 = 2
   Add 6: −y= 8
   Multiply by −1: y=− 8
   Plug in and solve for x: x=y− 2 =(−8) − 2 =− 10


5. (4, 3) and (2, −3) Substitute: x^2 −2(3x−9) = 10
   Distribute: x^2 − 6 x+ 18 = 10
   Subtract 10: x^2 − 6 x+ 8 = 0
   Factor: (x−4)(x−2) = 0
   (Look over Lesson 5 if that step was tough!)
   Zero Product Property: x=4 or x= 2
   Plug in and solve for y: y= 3 x− 9 =3(4) − 9 = 3
   or =3(2) − 9 =− 3
   So the solutions are x=4 and y=3 or
   x=2 and y=−3.


6. (2, 5) Substitute:

Simplify: 2 n+ 5 = 3 n

Subtract 2n: 5 =n

Plug in to find m: m= ()=

2

5

52

5

2

5

nn 53

⎛

⎝⎜

⎞

⎠⎟

+=

SAT Practice 2

1.C Substitute: 3 x+2(3x) = 72
Simplify: 9 x= 72
Divide by 9: x= 8

2.D Translate into equations: x−y= 4
x+y=− 7
Add the equations: 2 x =− 3
Divide by 2: x =−1.5
Substitute: −1.5 +y=− 7
Add 1.5: y=−5.5
(−1.5)(−5.5) =8.25

3. 6 Subtract them:
   2 m− 9 n=(4m− 7 n) −(2m+ 2 n)
   = 10 − 4 = 6

4.E Subtracting gives 2p=a+ 4
Divide by 2: p=(a+4)/2

5.C Translate: h+ 2 s= 5.40

3 h+s= 8.70

Multiply 2nd eq. by 2: 6 h+ 2 s=17.40

−(h+ 2 s= 5.40)

Subtract 1st equation: 5 h=12.00

Divide by 5: h= 2.40

6.A Divide the first equation by the second:

Simplify:

7.D Translate: x+y=5 and x−y= 2

Although you could solve this system by combin-

ing, it’s easier to remember the “difference of

squares” factoring formula:

x^2 −y^2 =(x+y)(x−y) =(5)(2) = 10

m

y yy

=× =

3618

23

m

m y

(^6) y
5

3 2

6

=÷