Method I

4x + 6y = 5 --- eqn (1)
2x + 4y = 3 --- eqn (2)

Multiply equation 2 by 2 to eliminate x

2x x 2 + 4y x 2 = 3 x 2
4x + 8y = 6 --- eqn (3)

Equation 3 - Equation 1

2y = 1

Divide both sides by 2

y = $\frac{1}{2}$

Subtitute y = $\frac{1}{2}$ in equation 1

4x + 6 x $\frac{1}{2}$ = 5

4x + 3 = 5
4x = 5 - 3
4x = 2

Divide both sides by 4

x = $\frac{2}{4}$

x = $\frac{1}{2}$

x + 2y = $\frac{1}{2}$ + 2 x $\frac{1}{2}$

x + 2y = $\frac{\mathrm{1\; +\; 2}}{2}$

x + 2y = $\frac{3}{2}$

x + 2y = 1$\frac{1}{2}$

Method II

Express 2x + 4y = 3 to be in the form (x + 2y) and make (x + 2y) the subject

2x + 4y = 3

Factorize 2 out.

2(x + 2y) = 3

Divide both sides by 2 to make (x + 2y) the subject.

(x + 2y) = $\frac{3}{2}$

Change the improper fraction to mixed fraction.

(x + 2y) = 1$\frac{1}{2}$

4x + 2y = 16 ----- eqn 1

6x -2y = 4 ----- eqn 2

Get rid of y. eqn 1 + eqn 2

10x + 0 = 20

10x = 20

Divide both sides by 10

x = $\frac{20}{10}$ = 2

Substitute x = 2 into ----- eqn 1

4 x 2 + 2y = 16

8 + 2y = 16

2y = 16 - 8

2y = 8

Divide both sides by 2.

y = $\frac{8}{2}$ = 4

y - x = 4 - 2 = 2

3x - 2y = 10 ----------- eqn 1

x + 3y = 7 ------------ eqn 2

In order to get rid of x, multiply ----- eqn 2 by 3

3x + 9y = 21 ----------- eqn 3

Subtract ---- eqn 1 from ---- eqn 3

9y - - 2y = 11

Note: - - = + or -(-) = +

9y + 2y = 11

11y = 11

Divide both sides by 11

y = $\frac{11}{11}$ = 1

Substitute y = 1 into ------- eqn 1.

3x - 2(1) = 10

3x - 2 = 10

3x = 10 + 2

3x = 12

Divide both sides by 3

x = $\frac{12}{3}$ = 4

3p = 4q ----- eqn 1

9p = 8q - 12 ---- eqn 2

From ---- eqn 1

Make p the subject by dividing both sides by 3.

p = $\frac{\mathrm{4q}}{3}$ ------- eqn 3

Substitute p into ---- eqn 2

9 x $\frac{\mathrm{4q}}{3}$ = 8q - 12

3 x 4q = 8q - 12

12q = 8q - 12

12q - 8q = - 12

4q = -12

Divide both sides by 4

q = $\frac{-12}{4}$ = -3

Substitute q = -3 into ---- eqn 3.

p = $\frac{\mathrm{4\; x\; -3}}{3}$

p = -4

pq = p x q

pq = -4 x -3

pq = 12