The problem consists of 7 sub-problems. The first four sub-problems ask to solve simple algebraic equations for $x$. The fifth sub-problem asks to find the relation between the height ($x$) and area ($y$) of a parallelogram with a base of 12 cm. The sixth sub-problem asks to find the area ($y$) when the height ($x$) is 5cm. The seventh sub-problem asks to find the height ($x$) when the area ($y$) is 90 $cm^2$.

1. Problem Description

The problem consists of 7 sub-problems. The first four sub-problems ask to solve simple algebraic equations for $x$. The fifth sub-problem asks to find the relation between the height ($x$) and area ($y$) of a parallelogram with a base of 12 cm. The sixth sub-problem asks to find the area ($y$) when the height ($x$) is 5cm. The seventh sub-problem asks to find the height ($x$) when the area ($y$) is 90 $cm^2$.

2. Solution Steps

(1) $x + 6 = 31$

Subtract 6 from both sides of the equation:

$x = 31 - 6$

$x = 25$

(2) $x - 42 = 20$

Add 42 to both sides of the equation:

$x = 20 + 42$

$x = 62$

(3) $x \times 7 = 56$

Divide both sides of the equation by 7:

$x = 56 / 7$

$x = 8$

(4) $x \div 5 = 9$

Multiply both sides of the equation by 5:

$x = 9 \times 5$

$x = 45$

(5) The area of a parallelogram is given by the formula:

$Area = base \times height$.

In this case, the base is 12 cm and the height is $x$ cm, and the area is $y cm^2$. So, the relation between $x$ and $y$ is:

$y = 12 \times x$

(6) Given $x = 5$ cm, we want to find the area $y$:

$y = 12 \times x$

$y = 12 \times 5$

$y = 60 cm^2$

(7) Given $y = 90 cm^2$, we want to find the height $x$:

$y = 12 \times x$

$90 = 12 \times x$

$x = 90 / 12$

$x = 7.5$ cm

3. Final Answer

(1) $x = 25$

(2) $x = 62$

(3) $x = 8$

(4) $x = 45$

(5) $y = 12x$

(6) $y = 60$ $cm^2$

(7) $x = 7.5$ cm