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Multiplication Property of Equality

Last Updated : 04 Apr, 2024
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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.

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:

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.

FAQs on Multiplication Property of Equality

Define multiplicative Property of equality.

The Multiplication Property of Equality is an equation’s of two sides that will stay equal if you multiply them both by the same non-zero value.

Can you multiply both sides of an equation by any number?

No, you can’t multiply both sides of an equation by any number because division by zero is undefinable and you cannot multiply an equation’s of both sides by zero.

Can you provide an example of how to use the Multiplication Property of Equality?

For instance, in the equation 3x = 9, you can use the Multiplication Property of Equality by dividing both sides by 3 to find that x = 3.

Is it necessary to use a non-zero number when applying the Multiplication Property of Equality?

Yes, it’s crucial to use a non-zero number to ensure that the equality remains valid. Multiplying both sides by zero would result in an undefined equation.



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