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Types of Equations



In the above table, we can see that:

  • 2x + 1 = 4 is a linear equation with one variable while x + 2 y – 3z = 5 is a linear equation with 3 variables.
  • x2 + 2x + 1 = 0 is a quadratic equation in ‘x’ and x2 + 2y2 – 3x = 5 is a quadratic equation in ‘x’ and ‘y’.
  • The third row contains examples of some cubic equations in 1, 2 and 3 variables.

The highest power (exponent) to which the variable in an equation is raised is known as the degree of the equation.

  • Linear equation is an equation of degree 1.
  • Quadratic equation is an equation of degree 2.
  • Cubic equation is an equation of degree 3.

Linear equations with one variable

An equation of the form ax + b = 0 where x is a variable, a and b are constants and a ≠ 0 is called a linear or simple equation.

 

The root or solution of the equation ax + b = 0 is given by

 

Description: 38650.png 

 

If we substitute this value of x in the above equation, the expression on the left hand side will be equal to the expression on the right. Hence, we can tell that Description: 43367.png satisfies the above equation.

 

Example: 6x + 12 = 0 is a linear equation with 1 variable.
The root of the above equation will be Description: 38644.png

 

Example
Find the value of x if Description: 38638.png
Solution
Given: Description: 38632.png
Description: 38625.png 

 

Example
Find the root of the equation Description: 38619.png
Solution
Given: Description: 38613.png
Description: 38607.png

 
Example
The number 50 is split into two parts such that the difference between them is 10. What is the value of smaller part?
Solution
Let us assume one of the parts be x.
The other part must be what will be left in 50 after taking x, i.e., 50 – x.
Given that the difference between the two parts is 10. Hence,
Description: 38601.png 
Description: 38595.png 
One part is x = 30.
So, the other part is Description: 38589.png Hence, the value of the smaller part is 20.

 

Example
In a 2 digit number, the digit in ten’s place is thrice the digit in unit’s place. If 36 is subtracted from the number, the digits are reversed. Find the number.
Solution
The digits are unknown; so let’s assume the unit’s place digit be ‘x’.
 
Now, as given in the question, digit in ten’s place is thrice which means three times the digit in unit’s place. Thus, ten’s place will be 3x.
 
When digits are known, the number can be written as:
 
10 × (digit in ten’s place) + 1 × (digit in unit’s place).
 
So, the original number will be Description: 38583.png 
 
When 36 is subtracted, digits are reversed which means the ten’s place will become x and unit’s place will become 3x.
 
The new number will be 10 × x + 3x = 13x.
 
From the given condition, 31x – 36 = 13x.
 
Description: 38576.png 
So, we found the value of x as 2, the original number will be 31 × 2 = 62.





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