Linear Equations in One Variable

The value of an algebraic expression depends on the values of the variables involved it. A polynomial in one variable whose degree is one is called a linear polynomial in one variable. When two expressions are separated by an equality sign, it is called an equation.

The general form of a linear equation in one variable is ax + b = 0, a ≠ 0, a and b are constants.

A number, which when substituted for the variable in the equation makes LHS equal to RHS, is called its solution.

To solve an equation, you can

  1. Add same number to both sides of the equation.
  2. Subtract same number from both sides of the equation.
  3. Multiply both sides of the equation by the same nonzero number.
  4. Divide both sides of the equation by the same non zero number.

Example: 5 + x = 8

Subtracting 5 from both sides of the equation, we get

5 + x - 5 = 8 - 5

x + 0 = 3

x = 3

So, x = 3 is the solution of the given equation.

Example: y - 2 = 7

Adding 2 to both sides of the equation, we get

y - 2 + 2 = 7 + 2

y = 9

Hence, y = 9 is the solution.

Example: 7x + 2 = 8

Subtracting 2 from both sides of the equation, we get

7x + 2 - 2 = 8 - 2

7x = 6

Dividing both sides by 7, we get

x = 6/7

Example: 2(x + 3) = 3(2x - 7)

The equation can be written as

2x + 6 = 6x - 21

6x - 21 = 2x + 6

6x - 21 + 21 = 2x + 6 + 21

6x = 2x +27

6x - 2x = 2x +27 - 2x

4x = 27

x = 4/27

Word Problems

Example: The present age of Jacob's father is three times that of Jacob. After 5 years, the difference of their ages will be 30 years. Find their present ages.

Let the present age of Jacob be x years.

Therefore, the present age of his father is 3x years.

After 5 years, the age of Jacob = (x + 5) years.

After 5 years, the age of his father = (3x + 5) years.

The difference of their ages = (3x + 5) - (x + 5) years, which is given to be 30 years.

3x + 5 - (x + 5) = 30

3x + 5 - x - 5 = 30

3x - x = 30

2x = 30

x = 15

Therefore, the present age of Jacob is 15 years and the present age of his father = 3x = 3 × 15 = 45 years.

Example: The sum of three consecutive even integers is 36. Find the integers.

Let the smallest integer be x.

Therefore, other two integers are x + 2 and x + 4.

Since, their sum is 36, we have

x + (x + 2) + (x + 4) = 36

3x + 6 = 36

3x = 36 – 6 = 30

x = 10

Therefore, the required integers are 10, 12 and 14.

Example: The length of a rectangle is 3 cm more than its breadth. If its perimeter is 34 cm find its length and breadth.

Let the breadth of rectangle be x cm.

Therefore, its length = x + 3

Now, since perimeter = 34 cm,

2(x + 3 + x) = 34

2x + 6 + 2x = 34

4x = 34 - 6

4x = 28

x = 7

Therefore, breadth = 7 cm, and length = 7 + 3 = 10 cm