## Equations

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# An Equation

When we solve any mathematical problem our answer must be in accordance with the conditions and requirements given in the problem otherwise our answer will not be true.

In algebra, we transform the verbal expressions into algebraic expressions (containing variables) but to solve the problem we must apply the given conditions on the expressions.
To find the value of the variable for given conditions we construct equations.

## Equality

In equation, there is a sign of an equality (=) and expressions on its both sides.
The equality shows that value of expressions on both sides are equal.
The equality must satisfy the conditions to obtain true results.

## LHS & RHS

LHS means Left Hand Side of an equation
RHS means Right Hand Side of an equation

∴ In an equation, expression on LHS must be equal to expression on RHS
9y + 3 = 75 is an equation,
∴ Value of 9y + 3 (LHS) must be equal to 75 (RHS)

If there is a sign other than the equality sign between LHS and RHS, it is not an equation.
9y + 3 > 72 is not an equation,
5x + 4 < 30 is not an equation.

## Solution of an Equation

In an equation, LHS = RHS is true for only a definite value of the variable in it.

The value of the variable which satisfies the equation is called the solution of the equation.

9y + 3 = 75 is valid for only y = 8
We can check it by putting y = 8 in the LHS,
LHS
= 9y + 3
= 9⋅8 + 3
= 72 + 3
= 75
= RHS
Thus, solution of the above equation is 8.
No other value of y would satisfy the above equation, we can check this by putting different value of y in the equation.

## Properties of The Equality

### ↓Class7

ℹ In a weight balance arm in horizontal position shows that weights in both pans are equal. If we add or remove some weight in one pan, same amount of weight must be added or removed in the other pan to maintain the balance otherwise the arm will tilt to the heavier side.

Equality can be imagined as weight balanced which will hold true only if we do the same mathematical operation on both sides of an equality.

● If we add the same number to both sides of an equality, it still holds.
5×4 = 16+4 is a number equation
LHS = RHS (=20)
Let’s add 5 on both sides of equality
(5×4)+5 = (16+4)+5
LHS=RHS (=25), equality is still valid after the operation.

● If we subtract the same number from both sides of an equality, it still holds.
5×4 = 16+4
LHS = RHS (=20)
Let’s subtract 5 from both sides of equality
(5×4)−5 = (16+4)−5
LHS = RHS (=15)

● If we multiply or divide both sides of the equality by the same non-zero number, it still holds.
5×4 = 16+4
LHS=RHS (=20)
Let’s multiply both sides of the equality by 5
(5×4)×5 =(16+4)×5
LHS = RHS (=100)

Let’s divide both sides of the equality by 5
(5×4)÷5 = (16+4)÷5
LHS = RHS (=4)

➧ An equality remains the same, when the expressions on the left and on the right are interchanged.
5×4 =16+4
LHS=RHS (=20)

Lets interchange the sides of equality
16+4 = 5×4
LHS=RHS (=20)

## Solving Equations

We can use the properties of equality to find the solution of the equation. We isolate the variable on one side of the equation then the number on the other side is it’s value.

Let’s find the value of 9y + 3 = 75
(We have to isolate y on the LHS which means we have to remove 3 and 9 from the LHS.)
9y + 3 = 75
(subtracting 3 from both sides will remove 3 from the LHS)
⇒ 9y + 3 − 3 = 75 − 3
⇒ 9y = 72
(dividing both sides by 9 will remove 9 from the LHS)
⇒ 9y ÷ 9 = 72 ÷ 9
y = 8
y = 8 is the solution of the given equation.

While solving an equation we should use the symbol of implies that‘ () or the word or between two equivalent equations.

### Transposing

Changing the side of a term or a factor from one side of the equation to the other side is called transposing.

#### Transposing Terms

➧ When a term is transposed it’s sign get changed.
9y + 3 = 75
transposing 3 from LHS to RHS
On transposing 3 become −3
⇒ 9y = 75 − 3
⇒ 9y = 72

Transposing a term is the same as adding or subtracting the number from both sides.

9y + 3 = 75
subtracting 3 from both sides
⇒ 9y + 3 − 3 = 75 − 3
⇒ 9y = 72

While transposing, (3 − 3) is omitted because it’s value is 0 and has no effect on the equation.

#### Transposing Factors

To transpose factors (constant or variable or terms), expressions on both side of equality must be in factor form.

➧ When a number is multiplied to one side transposed to the other side it divides the other side.
9y = 72
transposing the multiplier 9 from LHS to RHS
y = 72/9
y = 8

Transposing a multiplier is the same as dividing the number from both sides.

9y = 72
dividing both sides by 9
9y/9 = 72/9
y = 8

In transposing, 9/9 is omitted because they cancel each other out and has no effect on the equation.

➧ When a number divides one side transposed to the other side it is multiplied to the other side.
x/5 = 6
transposing the divisor 5 from LHS to RHS
x = 6 × 5
x = 30

Transposing a divisor is the same as multiplying the number to both sides.

x/5 = 6
multiplying both sides by 5
x/5 × 5 = 6 × 5
5x/5 = 6 × 5
x = 30

In transposing, 5/5 is omitted because they cancel each other out and has no effect on the equation.

➧ Variables can also be transposed like numbers from one side of the equation to the other side.
3x = 2x + 18
⇒ 3x − 2x = 18
(2x is transposed from RHS to LHS)
x = 18

## Problem Solving

➢ Assign a variable to the unknown (x or any other letter)
➢ If there is another unknown, it must be related to the first one in some way, write expression for it.
➢ Form expressions from the keywords in the problem
➢ Relate the expressions with each other to construct the equation
➢ Solve the equation for the solution.

#### (Ex 4.4)Solve the following:

2.(a) The teacher tells the class that the highest marks obtained by a student in her class is twice the lowest marks plus 7. The highest score is 87. What is the lowest score?

3.(ii) Laxmi’s father is 49 years old. He is 4 years older than three times Laxmi’s age. What is Laxmi’s age?

#### Ans

2(a) Let, the lowest mark be x
Then, the highest mark would be 2x + 7
According to question (A/Q)
Highest mark = 87
2x + 7 = 87
2x = 87− 7
2x = 80
x = 80/2
x = 40
∴ The lowest score = 40

3(ii) Let, Laxmi’s age be x years
Then, Laxmi’s father age would be 3x + 4 years
A/Q
3x + 4 = 49
⇒ 3x = 49−4
⇒ 3x = 45
x = 45/3
⇒ x = 15
∴ Laxmi’s age = 15 years