Algebraic Expressions & Equations：練習問題

A number processing machine takes an input \(x\), adds \(7\) to it, and then multiplies the result by \(2\) to produce an output. If the output of the machine is \(20\), what was the value of the input \(x\)?

Solve the following simultaneous linear equations for x and y:

\( 4x + 3y = 14 \quad \text{(1)} \)
\( 5x - 2y = 1 \quad \text{(2)} \)

What is the value of $$y$$ in the equation: $$\frac{4y}{5} - \frac{y-1}{2} = 3$$

Evaluate the expression \( 5p - q^2 + 3 \) when \( p = -2 \) and \( q = 4 \).

Simplify the algebraic expression \( F = \frac{3x^2 - 3x - 18}{x^2 - 9} \). Assuming the denominators are non-zero, find the value of \( x \) if the simplified expression \( F \) is subsequently equal to 4.

Consider the algebraic expression \( 7x^2 - 4x + 9 \). Identify the coefficient of the \( x \) term and the constant term.

If \( (2x + 1)(x - a) + b \equiv 2x^2 - 5x + 3 \) is an identity, find the values of the constants \( a \) and \( b \).

Given the formula \( V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3 \), make \( h \) the subject of the formula. Then, calculate the value of \( h \) when \( V = 12\pi \) and \( r = 2 \).

An algebraic expression is given by \( P = 4(x - 3) - 2(5 - 3x) \).

(a) Expand the brackets and simplify the expression for \( P \).
(b) Find the value of \( P \) when \( x = 2 \).
(c) If \( P = 18 \), solve the equation to find the value of \( x \).
(d) If a new expression \( Q = P + 2x - 5 \), find the simplified form of \( Q \) in terms of \( x \).

Machine P and Machine Q are used to print books.

a) When Machine P prints at a rate of \(x\) books per hour and Machine Q prints at a rate of \(y\) books per hour, Machine P works for 3 hours and Machine Q works for 2 hours, printing a total of 240 books. Formulate an algebraic equation relating \(x\) and \(y\).

b) In a different scenario, Machine P increases its printing rate by 5 books per hour, and Machine Q decreases its printing rate by 1 book per hour. If Machine P works for 2 hours and Machine Q works for 4 hours at these new rates, they print a total of 302 books. Formulate a second algebraic equation relating \(x\) and \(y\).

c) Solve the simultaneous equations obtained from parts (a) and (b) to find the values of \(x\) and \(y\).

d) An urgent order requires \((x+y)^2 - (x-y)^2\) books to be printed.

i) Using an appropriate algebraic identity, calculate the exact number of books required for this urgent order.

ii) If Machine P and Machine Q operate at their original rates (from part (c)) and work together, how many hours will it take them to complete this urgent order?