Simple Equations

Introduction

An equation is a statement of equality between two mathematical expressions containing one or more variables.

A lot of cases which involve real numbers and more complicated problems, the use of simple equations becomes very important. The following will help us in understanding how using simple equations can help in solving a problem.

For e.g.: The age of Rahul’s father is ten years older than thrice the present age of Rahul. What is Rahul’s age if his father’s age is 40?

Firstly a variable is assigned.

Let the present age of Rahul be “x” years.

Now, father’s age is ten years more than thrice of rahul’s age. So we get:

Father’s age = 10+ 3x

Father’s age is given as 40. Therefore we can equate and get:

10 + 3x = 40

Now this is called a simple equation. This equation can simplify problems and help us finding the age of Rahul’s father by just substituting Rahul’s age in place of x.

Equations

Using constants and variables, we form expressions and using expressions we get an equation.

Constants have fixed value. For e.g. 2,4, 100,4/5 etc. The word variable means something that can vary, i.e. change. A variable takes on different numerical values; its value is not fixed. Variables are denoted usually by letters of the alphabet, such as x, y, z, l, m, n, p etc.

From variables, we form expressions.

The expressions are formed by performing operations like addition, subtraction, multiplication and division on the variables. An expression is a mathematical phrase with numbers, variables and operators. For e.g. 2x-2, 2x+4, 4y etc.

From expressions we get equations. An equation is a condition on a variable such that two expressions in the variable should have equal value. It is a statement of equality between two mathematical expressions containing one or more variables.

For e.g. 2x + 4 =10, 4y =0 . This is an example of an equation since it has two expressions with an equality sign.

The left hand side (LHS) and right hand side(RHS) are equal in an equation

The value of the variable for which the equation is satisfied is called the solution of the equation.

Problem: 

1.

Substituting the value of x=3 in x+3 we get:

X+3= 3+3=6

But the right hand side (RHS) is 3. Hence LHS and RHS are not equal.

So the equation is not satisfied at x =3.

Substituting the value of x=0 in x+3 we get:

X+3= 0+3=3

But the right hand side (RHS) is 0. Hence LHS and RHS are not equal.

So the equation is not satisfied at x =0.

2. Check whether the value given in the brackets is a solution to the given equation or not: 7n + 5 = 19 (n = 2)

A value of the variable is a solution to the given equation if LHS=RHS when the value of variable is substituted in the equation.

Substituting n=2 in the equation we get LHS as:

7n +5 =7(2)+5 = 19

The value of LHS is 19 and the value of RHS is 19.

Since both the sides are equal, n=2 is a solution to the equation 7n+5 =19.

Setting up equations:

The rules for setting an equation are:

o The value that is not known or to be found is assigned a variable.

o Then the expressions are formed as per the conditions given in the question.

o Finally, the expressions are equated with values given in the question.

Example

The value that is not known or to be found is assigned a variable.

Let the number of chocolates that the boy initially had =x

Then the expressions are formed as per the conditions given in the question.

Number of chocolates given to the sister =2

Number of chocolates that the boy is left with = 2(given)

Number of chocolates that the boy is left with = Initial of number chocolates he had – no of chocolates given to the girl = x-2 =2

Finally, the expressions are equated with values given in the question.

Number of chocolates that the boy is left with:

x-3 =2

Therefore, the equation set up is x-3 = 2.

Solving an equation

Solving an equation implies that a value of the variable will be found such that LHS =RHS. Such a value of the variable is called solution of the equation.

In order to solve an equation, it is important to remember these points:

o If we add the same number to both sides of a balance equation, the balance is undisturbed.

o If we subtract the same number from both sides of a balance equation, the balance is undisturbed.

o If we multiply or divide both sides of the equation by the same number, the balance is undisturbed.

Thus if we fail to do the same mathematical operation on both sides of a balanced equation, the balance is disturbed.

These conclusions are also valid for equations with variables as, in each

Equation variable represents a number only.

Other than these following tips have to be kept in mind:

o Whenever we move a number from LHS to RHS or vice-versa the sign of that number will change. For e.g. if we transpose -5 from LHS to RHS or vice versa it will become +5.

o Similarly if we transpose +5 from LHS to RHS or vice versa it will become -5.

o If we transpose *5 to the other side, it will become /5.

o If we transpose /5 to other side, it will become *5.

Problem: 

Solve the following equations:

x – 1 = 0

Solution

Firstly, we will move to the other side. When we will do this, -1 will become +1. We get:

x-1=0

x=1

So, the solution to this equation is x =1.

o y + 4 = – 4

Firstly, we will move -4 to the other side. When we do this, it will become -4. W get:

y+4 = -4

y =-4 -4

y= -(4+4)

y= -8

So, the solution to this equation is y = -8.

3s + 12 = 0

Firstly, we will move +12 to the other side. When we do this, it will become -12. We get:

3s +12 =0

3s = -12

Now, when we will take *3 to the other side, it will become /3.

s = -12/3

s = -4

So the solution to this equation is -4.

Solutions to Equations

Just like we move from equations to solution by solving the equation to find the value of the variable, we can move from solution to the equation.

A solution in itself is a simplified version of the equation. Hence it is possible to create an equation from a solution. Importantly, it is possible to create multiple equations from a solution.

Problem: 

Construct 3 equations starting with x = 2

Solution:

Using the property that a mathematical operation on both sides of an equation leaves the equation undisturbed, we get:

X=2

Adding 2 on both sides, we get the first equation:

X+2 = 2+2

X+2 = 4.

Multiplying both sides of equation 1by 3, we get the second equation:

2(X+2) =8

Subtracting 5 from both sides of the original equation given , we get the third equation:

X-5 = 2-5

X-5= -3

Applications of Simple Equations to Practical Situations

Simple equations can be used to solve problems in real life. The method is first to form equations corresponding to such situations and then to solve those equations to give the solution to the problems.

Problem: 

1. Maya, Madhura and Mohsina are friends studying in the same class. In a class test in geography, Maya got 16 out of 25. Madhura got 20. Their average score was 19. How much did Mohsina score?

The value that is not known or has to be found is assigned a variable.

So, let the marks scored by Mohsina = x

Marks scored by Maya =16

Marks scored by Madhura =20

Total marks scored by all three = 16+20+x = 36 + x

Average marks scored by all three = 19

Average marks = total marks/3

Total marks/3= 19

Total marks =19*3 = 57

Total marks scored by all three = 16+20+x = 36 + x

36 + x =57

Transposing 36 will make it -36.

X = 57 -36

X =21

Hence the marks scored by Mohsina are 21 out of 25.

2. In an isosceles triangle, the base angles are equal. The vertex angle is 40°. What are the base angles of the triangle? (Remember, the sum of three angles of a triangle is 180°).

Solution:

In an isosceles triangle the base angles are equal.

Let the base angles be of x degrees each.

Sum of three angles of a triangle= 180°

40 + x +x = 180

40 + 2x = 180

Now, transposing 40 will make it -40.

2x = 180 -40

2x = 140

Now again transposing *2 will make it /2.

x=140/2

x=70

Thus the base angles are of 70 degrees each.

3. Ibenhal thinks of a number. If she adds 19 to it and divides the sum by 5, she will get 8.

Solution:

Let the number thought by Ibenhal = k.

Adding 19 to this and dividing the sum by 5 she gets 8.

(k+19)/5 =8

Transposing /5 will make it *5.

(k+19) = 40

Transposing +19 will make it -19.

k= 40-19

k= 21

hence the number thought of by  Ibenhal is 21.