Multiplication Property of Equality is when two sides of an equation are multiplied by the same value, and equality will remain true. It is one of the many properties of equality such as Addition, Subtraction, Division, Reflexive, Transitive, etc.
In this article, we will discuss the Multiplication Property of Equality in detail including its definition as well as converse.
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What is the Property of Equality?
Properties of Equality refers to the set of rules that maintain the balance and truth of an equation when operations like addition, subtraction, multiplication, or division are applied to both sides of the equation.
In other words, if a = b, then for any operation (such as addition, subtraction, multiplication, or division) performed on a, the same operation must be performed on b to preserve equality. The main properties of equality are:
- Addition Property
- Subtraction Property
- Division Property
- Multiplication Property
- Reflexive Property
- Transitive Property
- Substitution Property
- Symmetric Property
In this article, we will discuss the Multiplicative Property of Equality in detail.
What is the Multiplicative Property of Equality?
Multiplication Property of Equality states that if both sides of an equation are multiplied by the same number, equality holds. This property allows you to multiply both sides of an equation by the same number, maintaining the equality of the equation.
Property of equality is the qualities that do not affect the equality of two or more values or the truth value of an equation.
Multiplication Property of Equality: Definition
For real numbers a, b, and c, If a = b, then a × c = b × c.
This means an equation of two sides always stays equal when they are multiplied by the same real integer.
Converse of the Multiplication Property of Equality
For real numbers x, y and z; converse of multiplication property of equality can be expressed as:
If x ≠y, then x × z ≠y × z
OR
If x = y, then x × z = y × z
Multiplication Property of Equality with Fractions
We can apply multiplication property of equality with fractions as well, following examples can be seen for the same.
Let's say you have an equation:
a/b = c
To solve for a, you can multiply both sides of the equation by b:
a/b × b = c × b
This simplifies to:
a = bc
Similarly, if you have:
a/b ​= c/d​
You can solve for a by multiplying both sides by bd:
a/b ​× bd = c/d ​× bd
Which simplifies to:
ad = bc​
Conclusion
In conclusion, the Multiplication Property of Equality stands as a fundamental principle in algebra, facilitating the solution of equations by allowing us to balance both sides through multiplication by the same non-zero factor. Its simplicity belies its power, enabling us to manipulate equations with confidence and precision.
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Solved Examples: Multiplication Property of Equality
Example 1: Solve 3x = 12.
Solution:
Given: 3x​ = 12
Multiply both sides by 1/3
​3x × 1/3​ = 12 × 1/3
⇒ x = 12/3
⇒ x = 4
So, the solution for x is 4.
Example 2: If 3x/4 = 36, then find the value of x.
Solution:
Let's multiply both sides by the multiplicative inverse of 3/4 into 4/3.
3x/4 × 4/3 = 36 × 4/3
⇒ x = 48
Example 3: Let b and c be real numbers such that z/2 = 3x and z/3 = 2z. Then show how x = z.
Solution:
In First equation z/2 = 3
Let's multiply both sides by 2.
(z/2) × 2 = 3x × 2
⇒ z = 6x
In Second Equation z/3 = 2z, and Substitute the value of z also.
6x/3 = 2z
⇒ 2x = 2z
Practice Questions on Multiplication Property of Equality
Q1: Solve for x, 5x = 30.
Q2: If 3y = 27, Find the value of y.
Q3: Solve for y, (2/3)y = 18.
Q4: If 5(a - 1) = 25, what is the value of a.
Q5: Solve the equation; (1/5)x = 7, and determine the value of x.