Algebraic Expressions, Substitution and Rearranging：練習問題

Simplify the following expression by combining like terms:
\( 5a + 7b - 2a - 3b \)

Simplify the algebraic expression \( 5(2x - y) - 2(3x - 4y) \) by expanding the brackets and combining like terms.

The diagram below shows a sequence of patterns formed by connecting regular pentagons side by side. Figure 1 contains 1 pentagon, Figure 2 contains 2 pentagons, and Figure 3 contains 3 pentagons. If the number of line segments in Figure \( m \) is given by \( P(m) \), find an algebraic expression for the number of segments in Figure \( (2k + 1) \).

Evaluate the algebraic expression \( 2(a - 3b) + c \) given that \( a = 5 \), \( b = 1 \), and \( c = -4 \).

The diagram below shows a sequence of figures made from small circles. If the pattern continues, which of the following expressions represents the number of circles in Figure \( n \)?

Evaluate the algebraic expression \( 2(a + b)^2 - 3ab \) when \( a = 4 \) and \( b = -1 \).

Evaluate the value of the algebraic expression \( \frac{2a^2 - 3bc}{a + b} \) when \( a = 4 \), \( b = -2 \), and \( b = 3 \).

The perimeter of the triangle shown below is \( 40\text{ cm} \). Calculate the length of the longest side.

A student has several bags of marbles. Each bag contains \( n \) marbles. The student also has 6 extra marbles outside of the bags.

(a) Write an algebraic expression to represent the total number of marbles the student has.
(b) If the student gives away 2 bags of marbles, write a simplified algebraic expression for the number of marbles remaining.
(c) If each bag contains 15 marbles (i.e., \( n = 15 \)), calculate the total number of marbles remaining from the expression in part (b).

A sequence of shapes is constructed using matchsticks to form a row of regular hexagons, as shown in the provided diagram.
(a) Determine the number of matchsticks required to form Pattern 4.
(b) Find a general algebraic expression for the number of matchsticks, \( S \), required to form Pattern \( n \).
(c) Calculate the number of matchsticks needed for Pattern 25.
(d) If a student uses exactly 201 matchsticks, how many hexagons, \( n \), are in the pattern? Show your algebraic steps.