Résolution de problèmes — Algebra

1Problem 1 of 24

Equivalent expressions check. Determine whether each pair of expressions is equivalent, by expanding/simplifying:

(a)

3(x + 4)

and

3x + 12

.
(b)

(x + 2)^2

and

x^2 + 4

.
(c)

2(x - 3) - (x - 5)

and

x - 1

.
(d)

\dfrac{6x}{2}

and

3x

.

Working space

2Problem 2 of 24

Substitution with mixed signs. Calculate the value of each expression for

x = -3, y = 2

:

(a)

2x + 3y

(b)

x^2 - y^2

(c)

\dfrac{x + y}{x - y}

(d)

(2x - y)^2

Working space

3Problem 3 of 24

Factorise practice. Factorise each expression fully.

(a)

6x + 12

(b)

4x^2 - 8x

(c)

12 a^2 b - 18 a b^2

(d)

5(x - 1) + y(x - 1)

Working space

4Problem 4 of 24

Algebraic fractions. Simplify each:

(a)

\dfrac{8a^2 b}{4 a b^2}

(b)

\dfrac{6x^2 - 9x}{3x}

(c)

\dfrac{x}{2} + \dfrac{2x}{5}

(d)

\dfrac{x - 3}{4} - \dfrac{x + 1}{6}

Working space

5Problem 5 of 24

Songs in terms of $x$ (CdN Algebra Feb 2022). Pete has three times as many songs as Sabrina. Sabrina has 120 fewer songs than Elena.

(a) If Elena has

x

songs, write expressions for (i) Sabrina's songs and (ii) Pete's songs.
(b) Pete has 300 songs. Write and solve an equation to find Elena's count.

Working space

6Problem 6 of 24

Rectangle expressions. A rectangle has dimensions

(2x + 1)

and

(x + 3)

.

(a) Write expressions for perimeter and area.
(b) Expand the area expression.
(c) When

x = 4

, find numerical values for perimeter and area.

Working space

7Problem 7 of 24

Two-bracket expansion. Expand and simplify each:

(a)

(x + 5)(x + 3)

(b)

(x - 4)(x + 7)

(c)

(2x - 1)(x + 3)

(d)

(x + 2)^2

Working space

8Problem 8 of 24

Pete & Sabrina extended. Pete has 3 times as many songs as Sabrina. Sabrina has

d

fewer songs than Elena. If Pete has

P

songs, express Elena's number of songs in terms of

P

and

d

.

Working space

9Problem 9 of 24

Algebra in geometry. A triangle has angles

(2x)°

,

(3x + 10)°

,

(x - 10)°

.

(a) Set up an equation in

x

and solve.
(b) Find the three angles.

Working space

10Problem 10 of 24

Patterns in algebraic identities. Verify (by expansion) that for any

a

and

b

:

(a)

(a + b)^2 = a^2 + 2ab + b^2

(b)

(a - b)^2 = a^2 - 2ab + b^2

(c)

(a + b)(a - b) = a^2 - b^2

(d) Use (a) to compute

52^2

from

50^2 = 2500

.

Working space

11Problem 11 of 24

Comparing fractions algebraically. Simplify each and decide which is larger when

x > 0

:

(a)

\dfrac{x + 6}{2}

vs

\dfrac{2x + 6}{3}

(b) Find the value of

x

where they are equal.

Working space

12Problem 12 of 24

Repeating-decimal investigation. Recurring decimals can be turned into fractions algebraically. Let

x = 0.\overline{ab}

(two-digit repeating block).

(a) Show that

100x - x = ab

and hence

x = \tfrac{ab}{99}

.
(b) Use this to convert

0.\overline{36}

to a fraction in simplest form.
(c) Convert

0.\overline{142857}

— what fraction do you recognise?

Working space

13Problem 13 of 24

Linear-equation grand tour. Solve each, showing your steps:

(a)

4x + 7 = 31

(b)

3(x - 4) = 18

(c)

5x - 3 = 2x + 18

(d)

\dfrac{x + 3}{4} = 7

Working space

14Problem 14 of 24

Kite perimeter algebra. A kite has two long sides of length

(3n - 1)

cm and two short sides of length

(n + 4)

cm. The perimeter is 38 cm.

(a) Set up an equation in

n

and solve.
(b) Find the lengths.
(c) Verify the perimeter.

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15Problem 15 of 24

Songs equation. Pete has 3 times as many songs as Sabrina. Sabrina has 120 fewer than Elena. Pete has 300 songs.

Find Elena's count.

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16Problem 16 of 24

Rectangle algebra. In a rectangle, the long sides are

3y

and

(y + 3)

cm; the short sides are

z

and

(3z - 4)

cm.

(a) Solve for

y

.
(b) Solve for

z

.
(c) Find the perimeter.

Working space

17Problem 17 of 24

Reverse-percentage equation. A bike on sale for 590 chf is at a 26% discount.

(a) Set up an equation and find the original price.
(b) The shop raises the sale price by 26%. Show the new price is not the original, and find the difference.

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18Problem 18 of 24

Two-step word problem. I think of a number. I treble it, subtract 7, then halve the result. I get 4. What is my number?

Working space

19Problem 19 of 24

Pricing strategy. Tickets cost

x

chf each. A booking fee of 5 chf is added per order. A school orders 25 tickets and pays a total of 380 chf.

(a) Set up an equation and find the ticket price.
(b) If the booking fee is doubled, how does that change the per-ticket effective price for 25 tickets?

Working space

20Problem 20 of 24

The man and his son. A man is currently four times as old as his son. In four years' time, he will be three times as old as his son will be then.

(a) Let the son's current age be

s

. Write an equation modelling the situation.
(b) Solve and find the current ages.

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21Problem 21 of 24

Mixture problem. A container has 5 L of a mixture that is 20% acid. How many litres of pure water should be added to make a mixture that is 10% acid?

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22Problem 22 of 24

Equation from a triangle. A triangle has sides

(x + 3)

,

(2x - 1)

and

(x + 6)

cm. The perimeter is 36 cm.

(a) Set up and solve an equation.
(b) Find the three side lengths.
(c) Check the triangle inequality.

Working space

23Problem 23 of 24

Solving for $n$ in a value-per-bag context. Tea is sold in two packs. A small pack contains

n

tea bags for 3 chf; a large pack contains

5n - 4

tea bags for 11 chf.

(a) Write expressions for the cost per tea bag for each pack.
(b) For which

n

are the two packs equal value?
(c) For

n > 3

, which pack is better value?

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24Problem 24 of 24

Trial-and-improvement check. Show that the equation

x^2 + x = 30

has a solution between 5 and 6, and use a bisection-style search to find

x

to 1 d.p.

Working space

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