The Addition Property of Equality and the Subtraction Property of Equality can be used to solve equations. The example below illustrates the Addition Property of Equality.
14 = 14
14 + 3 =
14 + 3 Add
3 to each side.
17 = 17
Note that c can be
positive, negative, or 0. Look at the equations below. In the equation on the
left, c = 3. In the equation on the
right, c = –3.
a + c = b + c a + c = b + c
15 + 3 = 15 + 3 15
+ (–3) = 15 + (–3)
When the same number is added to each side of an equation,
the result is an equivalent equation. Equivalent equations
have the same solution.
x + 2 = 6 The solution of the equation is 4.
x + 2 + 3 = 6 + 3 Using the Addition Property of Equality,
add 3 to each side.
x + 5 = 9 The solution of this equation is
also 4.
The Addition Property of Equality can be modeled with algebra
tiles.
Model the equation x
– 3 = 5. Each side of the mat represents a side of the equation. When you add 3
tiles to each side, the result is an equivalent equation, x = 8.
Example
1. Solve each equation. Check your solution.
m – 10 = 14
Alternative Solutions :
m – 10 = 14 Original equation
m – 10 + 10 = 14 + 10 Add
10 to each side.
m + 0 = 24 –10
+ 10 = 0
m = 24 Simplify.
To
check that 24 is the solution, replace m in the original equation with 24.
Check: m – 10 = 14 Original equation
Replace m with 24.
14 = 14 ü 24 – 10 =
14
The
solution is 24.
2. –6.2 + x = 15.5
Alternative Solutions :
–6.2 +
x = 15.5 Original
equation
–6.2 +
x + 6.2 =
15.5 + 6.2 Add 6.2 to each side.
x + 0 = 21.7 Simplify.
x = 21.7 Simplify.
To
check that 21.7 is the solution, replace x
with 21.7.
Check:
–6.2 + x = 15.5 Original equation
Replace x with 21.7.
15.5 = 15.5 ü –6.2
+ 21.7 = 15.5
The
solution is 21.7.
In addition to the Addition Property of Equality, there is
also a Subtraction Property
of Equality.
Example
3. Solve 9 + a = –3.
Check your solution.
Alternative Solutions :
9 + a = –3 Original equation
9 + a – 9 = –3
– 9 Subtract
9 from each side.
a = –12 Simplify.
Check:
9 + a
= –3 Original equation
Replace a with –12.
–3 = –3 ü Add –12 to 9.
The
solution is –12.
Sometimes the Subtraction Property of Equality can be used
instead of the Addition Property of Equality. Recall that subtracting a number
is the same as adding its inverse.
Example
Art Link
4. Romanian Constantin Brancusi and American David Smith are successful
twentieth-century sculptors. Brancusi’s Bird
in Space and Smith’s Cubi XVIII were
created in 1918 and 1964, respectively. Use the equation 1918 + y = 1964 to find the number of years
between these two sculptures.
Alternative Solutions :
Method 1: Use the Subtraction
Property of Equality.
1918 +
y = 1964 Original equation
1918 +
y – 1918 =
1964 – 1918 Subtract
1918
y = 46 from
each side.
Check: 1918 + y = 1964 Original equation
Replace y with 46.
1964 = 1964 ü
Method 2: Use the Addition Property
of Equality.
1918 +
y = 1964 Original equation
1918 +
y + (–1918) =
1964 + (–1918) Add
–1918 to each side.
y = 46 The
answers are the same.
The
solution is 46. So, the sculptures were created 46 years apart.
Example
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