Solving Equations by Using Different Operations such as Multiplication and Division (Definition, Types and Examples) - BYJUS

# Solving Equations by Using Different Operations such as Multiplication and Division

An equation is a statement that states the equality of two expressions. Two expressions are connected using an “equal to” (=) sign to form an equation. We can solve for the unknown values in an equation by multiplying and dividing the terms on both sides by specific numbers. Keep these properties in mind to solve equations using simple math operations....Read MoreRead Less

## Solving Equations by Using Different Operations such as Multiplication and Division Equations:

In mathematics, an equation is a formula that connects two expressions with the equals sign (=) to express their equality.

Example: Suppose we have a seesaw in the park. Two children are playing on that seesaw. One child’s weight is 32 pounds, and the other’s is 16 pounds. So, the seesaw will tilt towards the heavy side, and if we have to balance it, we need two children weighing 16 pounds each to do this.

## Solving Equations by Using Different Operations such as Multiplication and Division

For some values, equations may be satisfied, while for others, they can be unsatisfied. A value that makes an equation satisfied is called a solution.

## The table below shows the correct equation formation:

Value of a

Equation

$$a\times 2=8$$

Satisfied/Unsatisfied

3

$$3\times 2=6$$

Unsatisfied

4

$$4\times 2=8$$

Satisfied

5

$$5\times 2=10$$

Unsatisfied

Hence, when the value of a = 4, it will form a unique solution for the equation a 2 = 8.

Opposite operations are referred to as inverse operations. In other words, they are operations that counteract the effects of other operations. Multiplication, for example, is the inverse of division, and addition is the inverse of subtraction.

## The multiplication property of equality

When both sides of an equation are multiplied by the same non-zero number, the two sides remain equal.

If a = b, then a . c = b . c

## The multiplicative inverse property

A non-zero number a and its reciprocal, $$\frac{1}{a}$$ , have a product of one.

$$n.~\frac{1}{n}~ = ~\frac{1}{n}. ~n~=~1,~ n~\neq~0$$

$$6.~\frac{1}{6}~ =~ 1$$

## The division property of equality

When both sides of an equation are divided by the same non-zero number, the two sides remain equal.

If  a = b, then  $$\frac{a}{c}~=~\frac{b}{c} , ~c~\neq~0$$

## Examples

Example 1:

Solve the following  2-step equations using multiplication and division operations.

(a)  $$(\frac{a}{4}) ~= ~-2$$

(b)  16 =  – 2a

Solution:

(a) We have $$(\frac{a}{4}) ~=~-2~(\frac{a}{4})~\times ~4~=~-2~\times 4$$   Using the multiplication property of equality

a =  – 8

The solution is a =  – 8.

(b) We have 16 =  – 2a $$\frac{16}{-2}~=~\frac{-2a}{-2}$$                         Using the division property of equality

a = -8

The solution is a = -8

Example 2:

Solve the following  2-step equations using reciprocals.

(a)  $$-~\frac{2}{3}a~=~-8$$

(b)  $$14~=~\frac{7}{8}a$$

Solution:

(a) We have  $$-\frac{2}{3}~=~-8$$

$$-\frac{2}{3}a~\times~ -~\frac{3}{2}~=~-8~\times ~-~\frac{3}{2}$$          Using the multiplication property of equality

a = 12

The solution is a = 12.

(b) We have $$\frac{7}{8}a$$.

$$14~\times ~\frac{8}{7}~=~\frac{7}{8}a~\times ~\frac{8}{7}$$                 Using the multiplication property of equality

a = 16

The solution is a = 16.

Example 3:  The temperature in the mega-mart is 62 degrees Fahrenheit. Each hour, the temperature drops by 6 degrees Fahrenheit. After how many hours will the temperature be 44 degrees Fahrenheit?

Solution:

In the mega-mart, the temperature is 62 degrees Fahrenheit.

The total change between the initial and final temperatures is 62 – 44 = 18 degrees Fahrenheit.

The hourly drop in temperature is 6 degrees Fahrenheit.

To find the time, let us form an equation and solve it.

Hourly change in temperature (°F per hour) Time (hour) = Change in Temperature (°F)

Suppose, the time required for falling of the temperature to 44° F is a.

So,

6 . a = 18

18 = 6a

$$\frac{18}{6}~=~\frac{6a}{6}$$       Using the division property of equality

a = 3

After 3 hours, the temperature will be 44° F.

The steps for using the subtraction and addition properties to solve equations are:

Step 1: Replace the variable in the equation with the number.

Step 2: Simplify both sides of the equation’s expressions.

Step 3: Check to see if the resulting equation is correct.

The number is a solution if it is true.

The steps to solve equations with one variable are:

Step 1: Using LCM, remove any fractions that may exist.

Step 2: Make both sides of the equation as simple as possible.

Step 3: Identify and isolate the variable.