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
- Add same number to both sides of the equation.
- Subtract same number from both sides of the equation.
- Multiply both sides of the equation by the same nonzero number.
- 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