Corrigé
8.8 Equations
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Solve $5x = 60$. | $x = 12$ |
| 2 | Solve $x + 7 = 20$. | $x = 13$ |
| 3 | Solve $x - 5 = 8$. | $x = 13$ |
| 4 | Solve $\dfrac{x}{4} = 7$. | $x = 28$ |
| 5 | A function machine: $x \to \times 3 \to + 4 \to $ output. If output is 19, find $x$. | $x = 5$ |
| 6 | Solve $3x + 5 = 14$. | $x = 3$ |
| 7 | Solve $10 - x = 4$. | $x = 6$ |
| 8 | Check whether $x = 5$ is a solution to $2x + 3 = 13$. | Yes |
| 9 | Solve $-x = -8$. | $x = 8$ |
| 10 | Solve $4x = -20$ where $rhs$ may be negative. | $x = -5$ |
| 11 | Solve $3x + 7 = 22$. | $x = 5$ |
| 12 | Solve $2(x + 3) = 14$. | $x = 4$ |
| 13 | Solve $5x - 5 = 2x + 10$. | $x = 5$ |
| 14 | Solve $\dfrac{x}{3} + 2 = 8$. | $x = 18$ |
| 15 | Solve $5(x - 4) = 25$. | $x = 9$ |
| 16 | Solve $\dfrac{2x + 3}{5} = 7$. | $x = 16$ |
| 17 | A kite has two pairs of sides $(2n + 1)$ cm and $(3n - 2)$ cm. The perimeter is 38 cm. Find $n$. | $n = 4$ |
| 18 | Solve $3x + 8 = -7$ when $c$ is negative. | $x = -5$ |
| 19 | I think of a number. I multiply by 4 and add 6. The result is 30. What is my number? | $x = 6$ |
| 20 | A rectangle has length $(x + 5)$ and width $(x - 1)$. Its perimeter is 28 cm. Find $x$. | $x = 6$ |
| 21 | Solve $\dfrac{3x - 5}{4} = 7$. | $x = 11$ |
| 22 | Solve $3(x - 4) = 2(5 + x)$. | $x = 22$ |
| 23 | Solve $\dfrac{x}{2} - \dfrac{x}{3} = 5$. | $x = 30$ |
| 24 | Pete has 3 times as many songs as Sabrina, who has 120 fewer than Elena. If Pete has 300 songs, how many does Elena have? | 220 |
| 25 | A student won 370 of 500 games, then won the next $x$ games. He has now won 75% overall. Find $x$. | $x = 20$ |
| 26 | Three consecutive integers sum to $96$. Find them. | 31, 32, 33 |
| 27 | Two angles in a triangle are $(2x + 10)°$ and $(3x - 20)°$, and the third is $90°$. Find $x$. | $x = 20$ |
| 28 | A square has perimeter $4(x + 3)$ cm. Find $x$ when the perimeter is 32 cm. | $x = 5$ |
| 29 | Solve $4x + 8 = 2x - 12$ where this gives a negative solution. | $x = -10$ |
| 30 | I think of a number. I double it and add 5. I then triple the result and subtract 2. The result is 28. Find my number. | $x = 0$ |
| 31 | Solve $\dfrac{x + 2}{3} = \dfrac{x - 5}{4}$. | $x = -23$ |
| 32 | Solve $\dfrac{2x + 1}{3} = \dfrac{x - 2}{2}$. | $x = -8$ |
| 33 | A laptop costs $x$ chf. The shop applies a 20% discount, then adds 7.7% VAT. The final price is 861.60 chf. Find $x$. | $x = 1000$ chf |
| 34 | Two pens and three pencils cost 5.40 chf. One pen and two pencils cost 3.20 chf. Find the cost of each item. | Pen 1.20 chf; pencil 1.00 chf |
| 35 | A man is 4 times his son's age now. In 4 years he will be 3 times. Find their current ages. | Son 8, man 32 |
| 36 | Solve $\dfrac{1}{x} + \dfrac{1}{2x} = \dfrac{3}{4}$. | $x = 2$ |
| 37 | Two consecutive integers have a product of 132. Find them by setting up and solving an equation. | 11 and 12 |
| 38 | A bottle holds 1.5 L of squash mix at concentration $x$% sugar. Adding 0.5 L of water reduces concentration to 9%. Find $x$. | $x = 12$ |
| 39 | Solve $5 - 2(3x - 4) = 7x - (x + 1)$. | $x = \tfrac{7}{6}$ |
| 40 | A rectangle has length $(2x + 1)$ and width $x$. Its area equals its perimeter. Find $x$. | $x = 3$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Solve $7x = 56$. | $x = 8$ |
| 2 | Solve $x + -3 = 12$. | $x = 15$ |
| 3 | Solve $x - 12 = -3$. | $x = 9$ |
| 4 | Solve $\dfrac{x}{6} = 5$. | $x = 30$ |
| 5 | A function machine: $x \to \times 3 \to + 4 \to $ output. If output is 19, find $x$. | $x = 5$ |
| 6 | Solve $4x + -3 = 13$. | $x = 4$ |
| 7 | Solve $25 - x = 8$. | $x = 17$ |
| 8 | Check whether $x = 5$ is a solution to $2x + 3 = 13$. | Yes |
| 9 | Solve $-x = -8$. | $x = -6$ |
| 10 | Solve $-3x = 12$ where $rhs$ may be negative. | $x = -4$ |
| 11 | Solve $5x + -4 = 21$. | $x = 5$ |
| 12 | Solve $4(x + -5) = 28$. | $x = 12$ |
| 13 | Solve $7x - 3 = 3x + 17$. | $x = 5$ |
| 14 | Solve $\dfrac{x}{5} + -1 = 4$. | $x = 25$ |
| 15 | Solve $7(x - 2) = 21$. | $x = 5$ |
| 16 | Solve $\dfrac{3x + -1}{4} = 5$. | $x = 7$ |
| 17 | A kite has two pairs of sides $(2n + 1)$ cm and $(3n - 2)$ cm. The perimeter is 28 cm. Find $n$. | $n = 3$ |
| 18 | Solve $5x + 12 = -3$ when $c$ is negative. | $x = -3$ |
| 19 | I think of a number. I multiply by 4 and add 6. The result is 30. What is my number? | $x = 5$ |
| 20 | A rectangle has length $(x + 5)$ and width $(x - 1)$. Its perimeter is 28 cm. Find $x$. | $x = 7$ |
| 21 | Solve $\dfrac{5x - 2}{3} = 11$. | $x = 7$ |
| 22 | Solve $5(x - 2) = 3(6 + x)$. | $x = 14$ |
| 23 | Solve $\dfrac{x}{3} - \dfrac{x}{4} = 2$. | $x = 24$ |
| 24 | Pete has 3 times as many songs as Sabrina, who has 120 fewer than Elena. If Pete has 360 songs, how many does Elena have? | 240 |
| 25 | A student won 300 of 500 games, then won the next $x$ games. He has now won 80% overall. Find $x$. | $x = 500$ |
| 26 | Three consecutive integers sum to $120$. Find them. | 39, 40, 41 |
| 27 | Two angles in a triangle are $(2x + 10)°$ and $(3x - 20)°$, and the third is $90°$. Find $x$. | $x = 10$ |
| 28 | A square has perimeter $4(x + 3)$ cm. Find $x$ when the perimeter is 32 cm. | $x = 7$ |
| 29 | Solve $5x + 6 = 3x - 4$ where this gives a negative solution. | $x = -5$ |
| 30 | I think of a number. I double it and add 5. I then triple the result and subtract 2. The result is 28. Find my number. | Same — 0 |
| 31 | Solve $\dfrac{x + 1}{2} = \dfrac{x - 3}{5}$. | $x = -11$ |
| 32 | Solve $\dfrac{2x + 1}{3} = \dfrac{x - 2}{2}$. | $x = 11$ |
| 33 | A laptop costs $x$ chf. The shop applies a 25% discount, then adds 7.7% VAT. The final price is 807.75 chf. Find $x$. | $x = 1000$ chf |
| 34 | Two pens and three pencils cost 4.50 chf. One pen and two pencils cost 2.70 chf. Find the cost of each item. | Pen 0.90 chf; pencil 0.90 chf |
| 35 | A man is 4 times his son's age now. In 4 years he will be 3 times. Find their current ages. | Son 6, man 30 |
| 36 | Solve $\dfrac{1}{x} + \dfrac{1}{2x} = \dfrac{3}{4}$. | $x = 8$ |
| 37 | Two consecutive integers have a product of 132. Find them by setting up and solving an equation. | 12 and 13 |
| 38 | A bottle holds 1.5 L of squash mix at concentration $x$% sugar. Adding 0.5 L of water reduces concentration to 9%. Find $x$. | $x = 18$ |
| 39 | Solve $5 - 2(3x - 4) = 7x - (x + 1)$. | $x = 2$ |
| 40 | A rectangle has length $(2x + 1)$ and width $x$. Its area equals its perimeter. Find $x$. | $x = 4$ |
Problèmes — Solutions détaillées
**Linear-equation grand tour.** Solve each, showing your steps: (a) $4x + 7 = 31$ (b) $3(x - 4) = 18$ (c) $5x - 3 = 2x + 18$ (d) $\dfrac{x + 3}{4} = 7$
(a) $x = 6$ (b) $x = 10$ (c) $x = 7$ (d) $x = 25$
**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.
(a) $8n + 6 = 38, n = 4$ (b) Long 11 cm, short 8 cm (c) $2(11) + 2(8) = 38$ ✓
**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.
$x = 220$
**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.
(a) $y = 3/2$ (b) $z = 2$ (c) 13 cm
**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.
(a) 797.30 chf (b) New price 743.40; difference 53.90 chf
**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?
Number = 5
**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?
(a) $x = 15$ chf (b) Effective rises from 15.20 to 15.40 chf/ticket
**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.
(a) $4s + 4 = 3(s + 4)$ (b) Son 8, man 32
**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?
5 L
**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.
(a) $4x + 8 = 36, x = 7$ (b) 10, 13, 13 cm (c) $10 + 13 > 13$ ✓ — valid
**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?
(a) Small: $3/n$; Large: $11/(5n - 4)$ (b) $n = 3$ (c) Large
**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.
$x \approx 5.0$ (exact: 5)