Balancing an Equation

What is an Equation?

An Equation is a statement showing that two algebraic expressions are equal in value.

2x + 3 = 6 – 9x is an example of an equation with an unknown x.

The equals sign means the expression on the left-hand side (LHS) is equal to the expression on the right-hand side (RHS).

2x + 3 = 6 – 9x is a linear equation as the highest power of x is 1.

Balancing an Equation:

Note that an equation will remain balanced provided what is done to one side is also done to the other side.

Example 1.1.1:

Solve the following equations

(i) 5y + 2 = 3y + 9
(ii) 9 – 4x = 2x – 3
(iv) 3 – x = 9 – 3x
(v) 12y – 7 + 2y = 2y – 3

Solution:

(i) 5y + 2 = 3y + 9

Subtract 3y from both sides

i.e. 5y – 3y + 2 = 3y – 3y + 9

⇒ 2y + 2 = 9

Subtract 2 from both sides

i.e. 2y + 2 – 2 = 9 – 2

⇒ 2y = 7

Divide both sides by 2

⇒ \( \frac {2y}{2} = \frac {7}{2} \\ \frac {\not{2}y}{\not{2}} = \frac {7}{2}\)

⇒ \(\scriptsize y = 3 \normalsize \frac{1}{2} \)

(ii) 9 – 4x = 2x – 3

Collect like terms

i.e. 9 + 3 = 2x + 4x

⇒ 12 = 6x

Divide both sides by 6

⇒ \( \frac {12}{6} = \frac {6x}{6} \)

⇒ \( \frac {12}{6} = \frac {\not{6}x}{\not{6}} \)

⇒ 2 = x

∴ x = 2

(iii) 3 – x = 9 – 3x

Add x to both sides

⇒ 3 – x + x = 9 – 3x + x

3 = 9 – 2x

Subtract 9 from both sides

⇒ 3 – 9 = 9 – 9 – 2x

⇒ -6 = -2x

Multiply both sides by -1

⇒ \( \scriptsize -6 \: \times \: -1 = \: -2x \: \times \: -1 \)

⇒ \( \scriptsize 6 = 2x \)

Divide both sides by 2

⇒ \( \frac {6}{2} = \frac {\not{2}x}{\not{2}} \)

⇒ 3 = x

∴ x = 3

(iv) 12y – 7 + 2y = 2y – 3

Re-arrange 

⇒ 12y + 2y – 7 = 2y – 3

⇒ 14y – 7 = 2y – 3

Collect like terms

⇒ 14y – 2y = -3 + 7 

⇒ 12y = 4

Divide both sides by 12

⇒ \( \frac {12y}{12} = \frac {4}{12} \)

⇒ y = \( \frac {4}{12} \)

⇒ y = \( \frac {1}{3} \)