Laws of Equality and properties are explained in detail here. Refer to all the properties, definitions, and examples in the below sections. Get the complete details of operations and formulae involved in it. Know the advantages and properties of equality laws to simplify, solve and balance the equations. Follow the steps to simplify the equations and get the solutions in an easy manner.
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Equality Properties
The equivalent equations are nothing but the equations which have the same solution. These equations are shown with the “=” sign which is termed as equality sign. In the below sections, we can find the statement of equality between two numbers, two mathematical expressions, a number, and a mathematical expression. The properties of equalities are as follows.
Reflexive Law
The reflexive property states that for all a belongs to R, “a” is equal to “a” i.e., a ∈ R, a = a where a is the real number
Example:
a = 2, then the reflexive property of a is 2 = 2
for a = √5, the reflexive property of a is √5 = √5
Symmetric Property of Equality
Symmetric means the uniformity, a regularity of agreement between something. In the symmetric property, there is equality or agreement of correspondence between something.
The symmetric property states that for all a, b belongs to R, “a” is equal to “b” tends to “b” is equal to “a” i.e., a, b ∈ R, a=b ⇒ b=a where a and b are real numbers.
Suppose that you have a circle, we can see that the length of the circle is the same which is known as symmetry.
Example:
If x = 10, then the symmetric property of x is 10 = x
If y = 5x + 8, then the symmetric property of y is 5x + 8 = y
Transitive Property of Equality
If any condition applies between two consecutive numbers of any sequence then that condition would surely be applied between any two numbers taken in order. The transitive property states that for all a,b and c belongs to R, “a” is equal to “b” tends to “b” is equal to “c” tends to “c” is equal to “a” i.e., a, b, c ∈ R, a=b ∧ b=c ⇒ c=a.
Suppose that there are 3 kids, Ramu, Somu, and Ramesh and each has an equal weight of 40 kgs. Therefore, we can say that the weight of Ramu is equal to the weight of Somu, and the weight of Somu is equal to the weight of Ramesh. So, from the above two equations, we can assume that the weight of Ramesh is equal to the weight of Ramu.
Example:
The length of the one side of a triangle from a to b is 10cm
The length of the other side of a triangle b to c is 10cm
As the two sides of the triangle are 10cm and 10cm
As a to b is equal to b to c, then c to a is also equal to 10cm
Therefore, a, b, c ∈ R, a=b ∧ b=c ⇒ c=a
Additive Property of Equality
If we have an equation of equality and if we add some amount of number or the same amount of something to both sides of the equation then even after addition, the equation of equality still remains true.
The additive property states that for a,b and belongs to R then “a” is equal to “b” tends to a plus “c” is equal to “b” plus “c” i.e., a, b, c ∈ R, a=b ⇒ a+c = b+c where a, b, c are real numbers. Suppose that the “a” plus “0” is equal to “0 plus a” is equal to a.
Example:
3 + (-3) = 0
Multiplication Property of Equality
If we multiply both sides of the equation with the same number, then even after the multiplication of the equation, the result remains the same.
The multiplicative property states that a, b, c belongs to R then “a” is equal to “b” tends to “ac” is equal to “bc” i.e., a, b, c ∈ R, a = b ⇒ ac = bc. Suppose that x is equal t\o y, then zx is equal to zy.
Example:
x = 25, then x + 5 = 25 + 5
Cancellation Property w.r.t Addition
There is something to cancel out from an equation of equality, for all a, b, c belongs to R, “a” plus “c” is equal to “b” plus “c” tends to “a” is equal to “b” ie., a, b, c ∈ R, a+c = b+c ⇒ a=b. Suppose that 7/5 + x and 7/5 + y are two numbers, then x=y
Example:
4 + x and 4 + y, then the cancellation property implies x = y
Cancellation Property w.r.t Multiplication
There is something to cancel out from an equation of equality, for all a, b, c belongs to R, “a” plus “c” is equal to “b” plus “c” tends to “a” is equal to “b” ie., a, b, c ∈ R, a*c = b*c ⇒ a=b. Suppose that 8x and 8y are two numbers, then x=y
Example:
6x = 6y, then the cancellation property implies x = y
Examples on Properties of Equalities
Problem 1:
Determine whether x = 3/2 is a solution of 4x – 2 = 2x + 1
Solution:
As given in the question,
x = 3/2
The equation is 4x – 2 = 2x + 1
Substitute the value of x = 3/2 in the above equation,
4(3/2) – 2 = 2(3/2) + 1
6-2 = 3 + 1
4 = 4
Hence, the equation is simplified
Therefore, the equation 4x – 2 = 2x + 1 is proved
Problem 2:
Solve x – 5/8 = 3/4
Solution:
As given in the question,
The equation is x – 5/8 = 3/4
Add 5/8 on both sides
After substituting the values, we get the value as x = 11/8
Substitute the value of x = 11/8, in the above equation,
Therefore, we get
11/8 – 5/8 = 3/4
6/8 = 3/4
3/4 = 3/4
Hence, the equation is simplified
Therefore, the equation x – 5/8 = 3/4 is proved.
Problem 3:
Solve the equation 9x – 5 – 8x – 6 = 7?
Solution:
As given in the question,
9x – 5 – 8x – 6 = 7
x – 11 = 7
Add 11 on both sides
x – 11 + 11 = 7 + 11
x = 18
Therefore the value of x = 17
Hence, the final solution is x = 17