JUSTIFICATION FOR SOLVING EQUATIONS
SIMPLE LINEAR EQUATIONS
The following algebraic properties are commonly used in solving equations:
The addition property of equality: if a = b , then a + d = b + d .
The multiplication property of equality: if a = b , then d ! a = d ! b .
The additive inverse property: a + (!a) = 0 .
The multiplicative inverse property: if a ! 0 , then
1
a
! a = 1.
The additive identity property: a + 0 = a .
The multiplicative identity property: 1! a = a .
The distributive property: a(b + c) = a ! b + a ! c .
In formal mathematics, justification of each step in a process may be required.
Example 1 Solve the equation x + 6 = !4 and justify each step.
x + 6 = !4!!!!!!
x + 6 + (!6) = !4 + (!6)!
x + 0 = !10
x = !10!!
Example 2 Solve the equation !3x =
!3x =
1
6
! 13 (!3x) = ! 13 ( 16 )
1
1x = ! 18
1
x = ! 18
given
addition property of equality
additive inverse
additive identity
1
6
and justify each step.
given
multiplicative property of equality
multiplicative inverse
multiplicative identity
Example 3 Solve the equation 3x ! 6 = !1 and justify each step.
3x ! 6 = !1!!!!!!
3x ! 6 + (6) = !1 + (6)!
3x + 0 = 5
3x = 5
given
addition property of equality
additive inverses
additive identity
1 !(3x) = 1 (5)
3
3
5
1x = 3 !
x = 53
multiplication property of equality
multiplicative inverses
multiplicative identity
Problems
In problem 1 provide the justification for each step and in problems 2 through 9 solve the
equation and justify each step.
1.
2(x + 4) = !7
2x + 8 = !7
2x + 8 + !8 = !7 + !8
given
a. ________________
2x + 0 = !15
2x = !15
c. ________________
1
2
b. ________________
d. ________________
( 2x ) = ( !15 )
1
2
1x = !
x=!
15
2
15
2
e. ________________
f. ________________
= !7 12
g. ________________
2.
!3x = 10
3.
7 + y = !3
4.
x+
5.
3x + 2 = !7
2
3
6.
!9 =
7.
!
y=5
8.
!2x ! 6 = !7
9.
3 ( x ! 2 ) = !9
1
2
m+3
10.
2
3
!5 =
= 1 12
c! 4
3
Answers (Justifications for problems 2 through 9 may vary.)
1a. distributive prop.
1b. (+) prop. of equal. 1c. (+) inverses
1d. (+) identity
1e. (x) prop. of equal.
1f. (x) inverses
3.
–10
4.
! 103
–24
7.
! 15
2
8.
–11
5
6
1
2
1g. (x) identity
5.
–3
2.
6.
9.
–1
10.
SOLVING SYSTEMS USING ELIMINATION
The multiplication property of equality and the addition property of equality also justify why
multiplying one equation by a number and adding it to another equation (the elimination method)
is valid and does not change the solutions to the system of equation.
Consider this example: 2x + 3y = 7
x + 4y = 1
2x + 3y = 7!!!!!!2x + 3y =!!7!
x + 4y = 1!!!!!"2x " 8y = "2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"5y = 5
The multiplication property of equality allows
us to multiply the second equation by –2 and
keep an equivalent equation.
The addition property of equality allows us to
add !2x ! 8y to both sides of the top equation,
but on the right side we are adding the
equivalent expression, –2, instead.
We now can easily find y = !1 and use it to find x = 5 .
Problems
Solve each system of equations and justify the steps.
1.
x + 2y = !7
2x ! 3y = 0
2.
2x ! 3y = 6
x + y = !12
3.
5x + 2y = !4
3x + 7y = 15
4.
7x + 2y = !8
!4x + 8y = 0
Answers (Justifications may vary.)
1.
(–3, –2)
2.
(–6, –6)
3.
(–2, 3)
4.
(!1, ! 12 )