Simple equations

Meaning of Equation

A sentence or statement says something about numbers written in symbols. Consider, for example the statement: “I think of a number, I then triple it and add 4 to it”. We proceed as follows to write the statement in symbols. Let the number thought of be “r”, tripling it gives “3r”, and by adding 4 we get “3r + 4”.

If the result of this process is 16, then we have 3r + 4 = 16.

This symbolic expression is called a simple equation, with one unknown quantity, r. The expression contains two sides referred to as left hand side, 3r + 4, and the right hand side 16. The symbol ‘=’ is called the equality sign, which tells us that what is on the right-hand side is the same as what is on the left hand side.

Solving Simple Equations

Examples:

Solve the following equations:

(i) x + 8 = 12  (ii) 72 = 9x  (iii) ^x/₂ = 5  (iv) 6 = ¹/₅x  (v) x – 4 = 18

Solutions:

(i) x + 8 = 12

The unknown x is on the LHS

Since 8 is added to x on the LHS, then we subtract 8 from both sides.

x + 8 – 8 = 12 – 8

x + 0 = 4

Therefore, x = 4

(ii) 72 = 9x

The unknown x is on the RHS

Since x is multiplying x on the RHS, then we divide both sides by 9.

⁷²/ = /

8 = x

Therefore, x = 8

(iii) ^x/₂ = 5

Since x is divided by 2 on the LHS, then we multiply both sides by 2

^x/ ×  = 5 x 2

x = 5 × 2 = 10

Therefore, x = 10

(iv) 6 = ¹/₅x

Here, we multiply both sides by 5

6 × 5 = ^x/ ×

30 = x

Therefore, x = 30

(v) x – 4 = 18

add 4 to both sides

x – 4 + 4 = 18 + 4

x + 0 = 22

x = 22

CLASS ACTIVITY

Solve the following equations:

(i) 8 = 5x + 3

(ii) ¹/₁₀x = 3¹/₂

(iii) 2t + 3 = 14

(iv) 19 = 10 + 3x

(v) 9c + 11 = 65

Translation of Word Problems into Equations and Vice Versa

Example 1:                                    

Translate the following statements into mathematical algebraic equations.

(a) I think of a number n, then subtract 4; the result is 10.

(b) I think of a number x, then add 5; the result is 12.

(c) I think of a number y; then double it, add 5, the result is 11.

(d) Think of a certain number, divide it by 3 then double then add 12; the result is 18.

(e) The sum of two consecutive whole numbers is 17. What are these numbers?

Solution:

(a) n − 4 = 10

(b) x + 5 = 12

(c) 2y + 5 = 11

(d) ^r/₃ + 12 = 18

(e) n + (n + 1) = 17

CLASS ACTIVITY

Translate the following word sentences into mathematical statements:

1. One-third of a number is subtracted from half and the result is one.
2. 5 is added to a number and the result is 9.
3. Two fifth of a number is added to 9 and the result is 12.

Word Problems

Examples:

(i) Olu thinks of a number, 3 is added to it and the result is 8. What is the number?

Solution:

Let the number be x

x + 3 = 8

Subtract 3 from each side:

x +3 – 3 = 8 – 3

x = 5

(ii) The sum of two consecutive odd numbers is 32. Find the value of the numbers.

Solution:

Let the smaller odd number be x

The next odd number = x + 2

Sum of the two odd numbers is x + x + 2

Given condition states that x + x + 2 = 32

2x + 2 = 32

2x = 32 – 2

2x = 30

x = 30 ÷ 2

x = 15

Therefore, the two odd numbers are 15 and 17.

PRACTICE QUESTIONS

Solve the following equations:

(i) 27 = 3 + 4y

(ii) ^x/₃ + 3 = 5

(iii) Doris is 22 years older than her son Ade. The sum of their ages at present is 58 years. Find their ages.

(iv) The sides of a triangle are x, x + 1 and x + 3 units. If the perimeter is 25cm, find the length of the longest side.

(v) 5(x – 3) = 4(x – 2)

ASSIGNMENT

1. Paul is 12 years older than Richard. Four years ago, Paul was four times as old as Richard. Find their present ages.
2. The angles of a triangle are x⁰, (x⁰+ 20) and 2x⁰. Find the value of the smallest angle.
3. Simplify 7q + 10p – 3r + (−8q + 3p + 5r )
4. Find the sum of 5y²– 8x²+ 3xy, xy – x² – 2y² and 6x² + 2xy – y²
5. The average of three consecutive numbers is 5. Find the smallest of the numbers.
6. Subtract 6x³– 8x²+ 4x – 2 from 5 – 4x + 6x² – 8x³