CHAPTER 8 / ESSENTIAL ALGEBRA I SKILLS 307
Answer Key 2: Systems
- (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 - (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 - (−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
- There are many solutions. Here are a few:
(0, 8); (20, 0); (10, 4); (5, 6) - 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
- Any set of equations that are true at the same time.
- Substitution and combination.
- Plug the solutions back into the equations and
check that both equations are true. - (−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 - (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. - (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
- 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