211 
… 3.1‹Equations and the addition property
„‹Please solve the following equations by using the addition
property.
ƒS
ˆ‰x - 5 = 3ˆˆˆx + 7 = 4
ˆ‰x - 5 + 5 = 3 + 5ˆ‰ x + 7 + (-7) = 4 + (-7)
ˆ‰x + 0 = 8ˆˆˆx‹+ 0 = -3
ˆ‰x = 8ˆˆˆŠ x = -3
‚S
The addition property of equality simply states that you may add the
same number to both sides of an existing equation.‹When solving an
equation such as, x - 6 = 9, the goal is to get 'x' by itself on one
side of the equation or the other.‹The number on the opposite side is
then the solution to the original equation.‹The solution is the number
that makes a true sentence out of the original equation.

The addition principle can be used to isolate 'x'.‹If 6 is added to
both sides of the equation‹x - 6 = 9,‹you get‹x - 6 + 6 = 9 + 6. The
-6 and +6 cancel to zero since they are additive inverses.‹Thus
x + 0 = 15 is the solution.
 1
ˆˆˆ‰ Solve‹x - 7 = 3


‰A)‹4ˆŠ B)‹-4ˆŠ C)‹10ˆ d) † of ‡

ˆˆˆˆ x - 7 = 3
ˆˆˆˆ x - 7 + 7 = 3 + 7
ˆˆˆˆ x + 0 = 10
ˆˆˆˆ x = 10
€ C
 2
ˆˆˆ‰ Solve‹x + 8 = 5


‰A)‹-3ˆŠB)‹13ˆŠ C)‹-13ˆd) † of ‡

ˆˆˆˆ x + 8 = 15
ˆˆˆˆ x + 8 + (-8) = 5 + (-8)
ˆˆˆˆ x + 0 = -3
ˆˆˆˆ x = -3
€ A
 3
ˆˆˆŠSolve‹6x + 5 = 5x - 2


‰A)‹-3ˆŠB)‹-7ˆŠ C)‹10ˆ d) † of ‡

ˆˆˆˆ6x + 5 = 5x - 2
ˆˆˆˆ6x + (-5x) + 5 = 5x + (-5x) - 2
ˆˆˆˆx + 5 = -2
ˆˆˆˆx + 5 + (-5) = -2 + (-5)
ˆˆˆˆx = -7
€ B
 4
ˆˆˆ‹Solve‹-3y + 10 = -4y - 8


‰A)‹-8ˆŠB)‹10ˆŠ C)‹-18ˆd) † of ‡

ˆˆˆˆ-3y + 10 = -4y - 8
ˆˆˆˆ-3y + 4y + 10 = -4y + 4y - 8
ˆˆˆˆy + 10 = 0 - 8
ˆˆˆˆy + 10 - 10 = -8 - 10
ˆˆˆˆy = -18
€ C
 5ˆˆSolveŠ5‰1
ˆˆˆˆŠΔωz = Δωz - 7
ˆˆˆˆŠ4‰4
ˆˆˆŠ 3
‰A)‹-7ˆŠB)‹ΔωzˆŠ C)‹-1ˆ d) † of ‡
ˆˆˆŠ 4
‰ 5‰1ˆˆ5‰Ϊ‹1‹ΏŠ1‰Ϊ‹1‹Ώ
ˆ‰Δωz = Δωz - 7ˆ Δωz + ³- Δωz³ = Δωz + ³- Δωz³ - 7
ˆ‰4‰4ˆˆ4‰ΐ‹4‹ΩŠ4‰ΐ‹4‹Ω

ˆ‰4
ˆ‰Δωz = 0 - 7ˆ z = 0 - 7ˆ‹z = -7
ˆ‰4
€ A
 6

ˆˆ‰ Solve‹3a + 2 + 2a - 4a = 4 + 11


‰A)‹-6ˆŠB)‹13ˆŠ C)‹9ˆ‹d) † of ‡
ˆˆ‰3a + 2 + 2a - 4a = 4 + 11
ˆˆˆŠ (3a + 2a - 4a) + 2 = 4 + 11
ˆˆˆŠ (a) + 2 = 15
ˆˆˆŠ a + 2 - 2 = 15 -2
ˆˆˆŠ a + 0 = 13
ˆˆˆŠ a = 13
€ B
 7

ˆˆ‹Solve 6k + 3 - 14k + 2k - 8 = -7k + 4


‰A)‹4ˆŠ B)‹-6ˆŠ C)‹9ˆ‹d) † of ‡
ˆˆŠ 6k + 3 - 14k + 2k - 8 = -7k + 4
ˆˆˆŠ(6k - 14k + 2k) + (3 - 8) = -7k + 4
ˆˆˆŠ-6k + (-5) = -7k + 4
ˆˆˆŠ-6k + 7k + (-5) = -7k + 7k + 4
ˆˆˆŠk + (-5) = 0 + 4
ˆˆˆŠk + (-5) + 5 = 4 + 5
ˆˆˆŠk = 9
€ C
 8

ˆˆ‰ Solve‹-2(x + 3) + 3(x - 6) =4


‰A)‹28ˆŠB)‹-6ˆŠ C)‹14ˆ d) † of ‡
ˆˆ‰ -2(x + 3) + 3(x - 6) = 4
ˆˆˆ‰-2x - 6 + 3x - 18 = 4
ˆˆˆ‰(-2x + 3x) + (-6 - 18) = 4
ˆˆˆ‰x + (-24) = 4
ˆˆˆ‰x + (-24) + 24 = 4 + 24
ˆˆˆ‰x + 0 = 28
ˆˆˆ‰x = 28
€ A
 9

ˆˆ‰Solve‹-6(2k - 4) + (2 + 13k) = 5


‰A)‹14ˆŠB)‹-21ˆŠC)‹6ˆ‹d) † of ‡
ˆˆ‰ -6(2k - 4) + (2 + 13k) = 5
ˆˆˆ‰-12k + 24 + 2 + 13k = 5
ˆˆˆ‰(-12k + 13k) + (24 + 2) = 5
ˆˆˆ‰k + 26 = 5
ˆˆˆ‰k + 26 + (-26) = 5 + (-26)
ˆˆˆ‰k + 0 = -21
ˆˆˆ‰k = -21
€ B
„‹Please identify the following equations as conditional,
identity, or inconsistent.
ƒSŠx - 6 = 7ˆ‹x - 6 = x - 6ˆŠx + 3 = x + 4
ˆŠ x-6 + 6 = 7 + 6Šx+(-x)-6 = x+(-x)-6Š x+(-3)+3 = x+(-x)+4
ˆŠ x + 0 = 13ˆ 0 - 6 = 0 - 6ˆŠ0 + 3 = 0 + 4
ˆŠ x = 13ˆ‰-6 = -6ˆˆ‹3 = 4
ˆŠ conditionalˆidentityˆˆ inconsistent
ˆŠ solution is 13Š solution is allˆ no solution
ˆˆˆˆ real numbers
‚S
First degree equations that have one solution are called conditional
equations.‹There is just one number that makes the original equation
a true sentence.

Some equations though are true no matter what number you substitute in
for 'x'.‹These are called identities and the solution set is all of the
real numbers.

The last type of linear equation is false no matter what number you
substitute in for 'x'.‹This type is called inconsistent and has no
solution.

x + 6 = 9ˆˆ‰ x + 6 = x + 6ˆˆ x + 6 = x + 7
conditionalˆˆŠ identityˆˆ‰ inconsistent
 10
ˆˆŠWhich type of equation is x - 6 = 14?


A) conditional‰B)‹identity‰C) inconsistent‰D) † of ‡

ˆˆˆˆ‹x - 6 = 14
ˆˆˆˆ‹x - 6 + 6 = 14 + 6
ˆˆˆˆ‹x + 0 = 20
ˆˆˆˆ‹x = 20
ˆˆ‰one solution implies conditional
€ A
 11
ˆˆˆŠWhich type of equation is
ˆˆˆ 5k + 4 - 12k + 4k - 8 = -3k +2


A) conditional‰B)‹identity‰C) inconsistent‰D) † of ‡
ˆˆ‹5k + 4 - 12k + 4k - 8 = -3k + 2
ˆˆˆ (5k - 12k + 4k) + (4 - 8) = -3k + 2
ˆˆˆ -3k + (-4) = -3k + 2
ˆˆˆ -3k + 3k + (-4) = -3k + 3k + 2
ˆˆˆ 0 + (-4) = 0 + 2
ˆˆˆ -4 = 2
ˆˆˆ inconsistent, no solution
€ C
 12
ˆˆˆ‹Which type of equation is
ˆˆˆ‰ 2(x + 3) = 2x + 6


A) conditional‰B)‹identity‰C) inconsistent‰D) † of ‡
ˆˆˆ2(x + 3) = 2x + 6
ˆˆˆ‰ 2x + 6 = 2x + 6
ˆˆˆ‰ 2x - 2x + 6 = 2x - 2x + 6
ˆˆˆ‰ 0 + 6 = 0 + 6
ˆˆˆ‰ 6 = 6
ˆˆ identity, solution is all real numbers
€ B
 13
ˆˆŠSolve‹5(-3x + 2) = -6(x - 2) - 9x - 2


 A) 6‰ B) all real numbers‰C) no solution‰ D) † of ‡

ˆˆˆ‹5(-3x + 2) = -6(x - 2) - 9x - 2
ˆˆˆ -15x + 10 = -6x + 12 - 9x - 2
ˆˆˆ -15x + 10 = -15x + 10
ˆˆˆˆ 10 = 10
ˆˆˆ‹identity, all real numbers
€ B