Algebraic notation, manipulation and proof：練習問題

Solve the following equation to find the value of \(x\):
\(4x + 7 = 23\)

A rectangular field has a length of \( (2x + 3) \) metres and a width of \( (x - 1) \) metres.
Given that the area of the field is \( 42 \text{ m}^2 \), show that \( 2x^2 + x - 45 = 0 \) and find the value of \( x \).

Solve the following simultaneous equations:
\(2x + y = 7\)
\(3x - y = 8\)

Expand the following expression:
\(3(2x - 5)\)

Find the value of \(x\) in the following linear equation:
\(3(x - 4) = 15\)

Given that \(y = 4x + 3\), find the value of \(y\) when \(x = 6\).

Solve the following equation for \(x\):
\(7x - 2 = 3x + 10\)

Solve the following simultaneous equations:
\( 3x + 2y = 16 \)
\( 2x - y = 6 \)

(a) Factorise fully: \(6y^2 + 15y\)
(b) Given the formula \(v^2 = u^2 + 2as\), find the value of \(v\) when \(u = 5\), \(a = 3\) and \(s = 4\). Assume \(v > 0\).

(a) Expand and simplify: \((x + 6)(x - 2)\)
(b) Solve the quadratic equation \(x^2 + 4x - 12 = 0\) by factorising.