Name: _____________________________________________________ Date: __________________________ Period: ____________
6.1B KEY Solve Equations and Justify
You have discussed many algebraic properties and are now ready to logically justify your
reasons for solving equations. Justify each step of the following solution of an equation.
1.)
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
j.)
k.)
l.)
(𝟐𝒙 + 𝟑) + 𝟕 = 𝟓𝒙 + 𝟐
2𝑥 + (3 + 7) = 4𝑥 + 2
2𝑥 + 12 = 4𝑥 + 2
– 2x
– 2x
0 + 12 = 2𝑥 + 2
12 = 2𝑥 + 2
–2
–2
10 = 2x + 0
10 = 2x
2
2
5 = 1𝑥
5=𝑥
𝑥=5
𝟔𝒙+𝟒
2.)
𝟒
6𝑥+4
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
Associative Prop
Addition
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Multiplicative Inverse
Multiplicative Identity
Symmetric
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
j.)
k.)
l.)
m.)
Property of Equality
Multiplicative Inverse
Distributive Prop
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Multiplicative Inverse
Multiplicative Identity
Symmetric
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
Distributive Prop
Commutative Prop
Addition
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Multiplicative Inverse
Multiplicative Identity
= 𝟐𝒙 − 𝟓
a.) 4 ( 4 ) = 4(2𝑥 − 5)
b.) 6𝑥 + 4 = 4(2𝑥 − 5)
c.) 6𝑥 + 4 = 8𝑥 − 20
d.) −6𝑥
−6𝑥
e.)
0 + 4 = 2𝑥 − 20
f.)
4 = 2𝑥 − 20
g.)
+ 20
+ 20
h.)
24 = 2𝑥 + 0
i.)
24 = 2x
j.)
2
2
k.)
12 = 1𝑥
l.)
12 = 𝑥
m.)
𝑥 = 12
3.)
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
j.)
k.)
l.)
𝟑𝒙 + 𝟐(𝟒 + 𝒙) = 𝟏𝟑
3𝑥 + 8 + 2𝑥 = 13
3𝑥 + 2𝑥 + 8 = 13
5𝑥 + 8 = 13
–8
–8
5𝑥 + 0 = 5
5x = 5
5
5
1x = 1
𝑥=1
Fill in the missing step for solving each equation and the missing justifications.
4.)
a.)
b.)
c.)
d.)
e.)
f.)
g.)
𝟏𝟐(𝒕 − 𝟓) + 𝟓 = 𝟎
12𝑡 − 60 + 5 = 0
12𝑡 − 55 = 0
+ 55 +55
12𝑡 + 0 = 55
12t = 55
12
12
55
1𝑡 = 12
h.)
5.)
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
𝑡=
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
j.)
12
𝟗𝒙 − 𝟕 = 𝟐(𝟓𝒙 + 𝟏𝟐)
9𝑥 − 7 = 10𝑥 + 24
-9x
-9x
0 − 7 = 1𝑥 + 24
−7 = 1𝑥 + 24
– 24
– 24
−31 = 1𝑥 + 0
−31 = 1𝑥
−31 = 𝑥
𝑥 = −31
𝒙−𝟑
6.)
55
=
a.)
b.)
c.)
d.)
e.)
f.)
g.)
Distributive Prop.
Addition
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Multiplicative Inverse
h.)
Multiplicative Identity
a.)
b.)
c. )
d.)
e.)
f.)
g.)
h.)
i.)
Distributive Prop
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Additive Inverse
Additive Identity
Multiplicative Identity
Symmetric
a.)
b.)
c.)
d.)
e.)
f.)
g.)
h.)
i.)
j.)
Property of Equality
Multiplication Inverse
Distributive Prop
Multiplication
Property of Equality
Additive Inverse
Additive Identity
Property of Equality
Multiplicative Inverse
Multiplicative Identity
𝟐
𝟖
𝟓
𝑥−3
2
5 • 8 ( ) = 40 ( ) 5 • 8
8
5(𝑥 − 3) = 8(2)
5𝑥 − 15 = 8(2)
5𝑥 − 15 = 16
+ 15 +15
5𝑥 + 0 = 15
5x = 15
5
5
1𝑥 = 3
𝑥=3
5
Solve each equation and justify each step. Answers may vary.
7.) 3( x  4)  2(3x  2)  1
8.) 2x  25  5x  10
7
2
9.) 𝑥 = 6