Simple Equations: Suppose Ravi goes to his grandfather to ask him his age. His grandfather tells Ravi that if he multiplies his age by 6 and then adds 8 to the product, he will know the age of his grandfather. Ravi got 62 as an answer. What is the age of Ravi? How do you find that? Ravi is 9 years old. Suppose a basket contains some apples and oranges.
The number of oranges is thrice of the number of apples. The total number of fruits in the basket is 200. How do you find the number of apples and oranges? There are 50 apples and 150 oranges. These are the examples of simple equations. In this section, we will study simple equations and how to solve them.
Simple equations are the general way of representing a relationship between a set of variables or the unknowns. Why are they called equations? An equation is an equality of expressions. It involves the equality (=) sign. Simple equations are the conditions on the variables. An equation tells that the expression on the left side is equal to that on the right side. The expression on the left side of ‘=’ is the LHS and the one on the right side is the RHS.
Terminologies of Simple Equations
Let us make ourselves familiar with some of the terms. A simple equation has two parts – Variables and Constants.
- Variables: As the name suggests variables means ‘vary or changeable’. Variables denote the unknown values. They are represented by English letters like a, b, c, x, y, z etc. They can take any value.
- Constants: Constants have the fixed numerical value like 1, 2, 5, 10, 87 etc. These are unchangeable.
- Root or Solution: The value of the variable for which LHS = RHS is the root or the solution of a simple equation.
- Terms: The parts of a simple equation joined by + or − sign are terms.
Converting Statement into Equation
Consider on your birthday you want to give your classmates chocolates. There are 40 students in your class other than you and you have 87 chocolates. If you give them two chocolates each, how many do you still have?
Let’s convert this statement into a mathematical equation. Let the number of remaining chocolates be x. You have given two chocolates to each student. Therefore, the number of chocolates distributed is 2 × 40 = 80.
Number of remaining chocolates = Number of total chocolates − Number of distributed chocolates.
⇒ x = 87 − 80 or, x = 7.
Converting Equation into Statement
Consider an equation, 3x − 5 = 17. The possible statement for this equation is 5 taken away from thrice of a number gives 17.
Solving an Equation
Solving an equation means to find the solution of the given equation. A simple equation has an equal (=) sign. This equality shows the balance of both the sides of the equation. The equation remains the same when the expressions are interchanged. There are some methods to solve simple equations.
Trial and Error Method
Substituting different values of the variable and checking the equality of LHS and RHS is the trial and error method. Let us solve the equation 3x + 5 = 17. We start to substitute different values of x. The value for which both the sides are balanced is the required solution.
|x||LHS = 3x + 5||RHS = 17|
|1||3x + 5 = 3×1 + 5 = 8||17|
|2||3x + 5 = 3×2 + 5 =11||17|
|3||3x + 5 = 3×3 + 5 = 14||17|
|4||3x + 5 = 3×4 + 5 = 17||17|
For x = 4, LHS = RHS. The required solution is x = 4.
Method of Balancing
The equality of both the sides of an equation shows the balance between the two. This balance remains the same even if we add, subtract, multiply or divide both sides of the equation by the same number. Suppose we have to solve, x − 3 = 15. Add 3 to both sides of the equation.
⇒ x − 3 + 3 = 15 + 3 or, x = 18.
Method of Transposing
Transposing means changing the sides of the number in the equation. If a positive number is transposed, it becomes a negative number and vice-versa. The sign of the number gets changed. The basic operators also get changed.
Addition becomes subtraction and vice–versa and the multiplication will become the division and vice–versa. Consider we have an equation, 5x + 14 = 244. We shift the numbers of LHS to RHS.
⇒ 5x = 244 − 14 = 230 or, x = 230 ÷ 5 = 46.
Solved Examples for You
Question: Write the equations for the following statements:
- Two added to the product of a number and 13.
- Divide the sum of one-fifth of a number and 6 by 7.
- Let the number be x. The equation is (13 × x) + 2 = 13x + 2.
- Let the number be a. The equation is (6 + a/5) ÷ 7.
Question: Solve 2x + 16 = 26 by trial and error method.
|x||LHS = 2x + 16||RHS = 26|
|1||2x + 16 = 2×1 + 16 = 18||26|
|2||2x + 16 = 2×2 + 16 = 20||26|
|3||2x + 16 = 2×3 + 16 = 22||26|
|4||2x + 16 = 2×4 + 16 = 24||26|
|5||2x + 16 = 2×5 + 16 = 26||26|
Here, LHS = RHS for x = 5 is the solution.