Simple Equation
(Chapter 4 in VII CBSE Maths)
An equation is a condition on a variable. The word variable means something that can vary (i.e. change its value from time to time). 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.
Forming an Expression:
From variables, we form expressions. The expressions are formed by performing operations like addition, subtraction, multiplication and division on the variables. For instance, from variable x, we can form the expression (4x + 5). For this, first we multiplied x by 4 and then added 5 to the product. Similarly, from y, we form the expression (10y – 20). For this, we multiplied y by 10 and then subtracted 20 from the product.
Evaluating an Expression:
The value of an expression thus formed depends upon the chosen value of the variable. As we have already seen,
when x = 1, 4x + 5 = 9;
when x = 5, 4x + 5 = 25.
Similarly,
when x = 15, 4 x + 5 = 4×15 + 5 = 65;
when x = 0, 4 x + 5 = 4 × 0 + 5 = 5
Solution of an Equation:
Equation 4x + 5 = 65 is a condition on the variable x. It states that the value of the expression (4x + 5) is 65.
Here, the condition is satisfied when x = 15. It is the solution to the equation 4x + 5 = 65.
When x = 5, 4x + 5 = 25 and not 65. Thus x = 5 is not a solution to the equation. Similarly, x = 0 is not a solution to the equation.
No other value of x other than 15 satisfies the condition 4x + 5 = 65.
Components of an Equation:
In an equation there is always an equality sign. The equality sign shows that the value of the expression to the left of the sign (the left hand side or L.H.S.) is equal to the value of the expression to the right of the sign (the right hand side or R.H.S.).
In equation 10y – 20 = 50, the L.H.S. is (10y – 20) and the R.H.S. is 50.
If there is some sign other than the equality sign between the L.H.S. and the R.H.S., it is not an equation. Thus, 4x + 5 > 65 is not an equation.
In equations, we often find that the R.H.S. is just a number. But this need not be always so. The R.H.S. of an equation may be an expression containing the variable.
For example, the equation
4x + 5 = 6x – 25
has the expression (4x + 5) on the left and (6x – 25) on the right of the equality sign.
In short, an equation is a condition on a variable. The condition is that two expressions should have equal value. Note that at least one of the two expressions must contain the variable. An equation remains the same, when the expression on the left and on the right are interchanged.
Exercises:
1. Write the following statements in the form of equations:
(i) The sum of three times x and 11 is 32.
(ii) If you subtract 5 from 6 times a number, you get 7.
(iii) One fourth of m is 3 more than 7.
(iv) One third of a number plus 5 is 8.
2. Convert the following equations in statement form:
(i) x – 5 = 9
(ii) 5p = 20
(iii) 3n + 7 = 1
(iv) m/5 – 2 = 6
3. Check whether the value given in the brackets is a solution to the given equation or not:
(a) n + 5 = 19 (n = 1) (b) 7n + 5 = 19 (n = – 2) (c) 7n + 5 = 19 (n = 2)
(d) 4p – 3 = 13 (p = 1) (e) 4p – 3 = 13 (p = – 4) (f) 4p – 3 = 13 (p = 0)
4. Solve the following equations by trial and error method:
(i) 5p + 2 = 17 (ii) 3m – 14 = 4
5. Write equations for the following statements:
(i) The sum of numbers x and 4 is 9. (ii) The difference between y and 2 is 8.
(iii) Ten times a is 70. (iv) The number b divided by 5 gives 6. (v) Three fourth of t is 15.
(vi) Seven times m plus 7 gets you 77. (vii) One fourth of a number minus 4 gives 4.
(viii) If you take away 6 from 6 times y, you get 60. (ix) If you add 3 to one third of z, you get 30.
6. Write the following equations in statement forms:
7. Raju’s father’s age is 5 years more than three times Raju’s age. Raju’s father is 44 years old. Set up an equation to find Raju’s age
8. A shopkeeper sells mangoes in two types of boxes, one small and one large. A large box contains as many as 8 small boxes plus 4 loose mangoes. Set up an equation which gives the number of mangoes in each small box. The number of mangoes in a large box is given to be 100.
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