Corrigé
7.3 Introduction to Algebra
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | If $a = 4$, find a value of $b$ so that $3a + b = 15$. Justify your answer. | $b = 3$; because $3 \times 4 = 12$ and $12 + 3 = 15$ ✓ |
| 2 | Which of these is the correct algebraic convention: $n3$ or $3n$? Explain why the other form is wrong. | $3n$ is correct; in algebra the coefficient is written before the variable |
| 3 | Write $$8x$ in two different ways as a sum of like terms. Show each is equivalent to the original. | E.g. $5x + 3x$ and $4x + 4x$; both simplify to $8x$ ✓ |
| 4 | Make up an equation whose solution is $x = 8$. Solve it to check. | E.g. $x + 4 = 12$; solving: $x = 12 − 4 = 8$ ✓ |
| 5 | A number of chairs $n$ is arranged in 4 equal rows. Write an equation for this and solve it to find $n$. | $4n = 28$... wait — $n = 7$ chairs per row; equation from context: "total = rows × per row" |
| 6 | Expand $3(x + 4)$ and show that your answer gives the same value as the original expression when $x = 2$. | $3x + 12$; when $x = 2$: bracket gives $3(6) = 18$; expanded gives $6 + 12 = 18$ ✓ |
| 7 | Solve $2x + 5 = 17$ and check your answer by substituting back. | $x = 6$; check: $2(6) + 5 = 17$ ✓ |
| 8 | Write two different expressions that both describe "a number $n$ multiplied by 4, then subtract 3". Explain why they are the same. | $4n - 3$ and $4 \times n - 3$ are the same; in algebra, $4n$ means $4 \times n$ |
| 9 | Decide: are $9y$ and $4y$ like terms? Simplify $9y - 4y$ and justify your step. | Yes, both have $y$; $9y - 4y = 5y$ because $9 - 4 = 5$ |
| 10 | In the expression $7x + 3$, identify the coefficient of $x$ and the constant. Explain what each part tells you about the expression. | Coefficient of $x$ is 7 (multiplies $x$); constant is 3 (never changes) |
| 11 | Little puzzle: what number $a$ makes $a \times 5 = 0$? How did you know? | $a = 0$. Multiplying anything by $0$ gives $0$, and $5$ on its own is not zero, so $a$ has to be the $0$. |
| 12 | Find the value of $2a + 3b$ when $a = 3$ and $b = 4$. | 18 |
| 13 | Simplify: $5x + 3y + 2x - 1y$. | $7x + 2y$ |
| 14 | Expand and simplify: $4(x + 3) + 5$. | $4x + 17$ |
| 15 | Solve: $4x - 3 = 21$. | $x = 6$ |
| 16 | Tickets to a show cost $\pounds {p}$ each. 5 friends buy tickets. Write an expression for the total cost, then find the cost when $p = 8$. | Expression: $5p$; cost = £40 |
| 17 | Find the value of $3x^2$ when $x = 4$. | 48 |
| 18 | Simplify: $6a + 5b + 2a - 3b$. | $8a + 2b$ |
| 19 | A rectangle has length $(x + 3)$ cm and width $4$ cm. Write a simplified expression for its perimeter. | $(2x + 14)$ cm |
| 20 | Expand the bracket: $4(2x - 5)$. | $8x - 20$ |
| 21 | Solve: $\dfrac{x}{3} + 4 = 9$. | $x = 15$ |
| 22 | Another little puzzle: what number $a$ makes $3(4 - a) = 0$? Try to spot the answer without doing lots of working. | $a = 4$. The $3$ isn't zero, so whatever's in the bracket has to be the zero — meaning $4 - a = 0$, so $a = 4$. |
| 23 | Simplify: $3x + 5x - 2x$. | $6x$ |
| 24 | Find the value of $3a^2 + 2a$ when $a = -2$. | 8 |
| 25 | Expand and simplify: $3(x + 2) + 2(x + 5)$. | $5x + 16$ |
| 26 | Solve: $5x + 12 = 2$. | $x = -2$ |
| 27 | A bag of apples costs ${p}p$ pence and a bag of oranges costs ${q}p$ pence. Write and simplify an expression for the total cost of 3 bags of apples and 4 bags of oranges. | $3p + 4q$ pence (cannot simplify further as $p$ and $q$ are different) |
| 28 | Work out $4a - 3b$ when $a = -3$ and $b = 2$. | $-18$ |
| 29 | Solve: $5x + 3 = 2x + 15$. | $x = 4$ |
| 30 | A rectangle has length $(2x + 3)$ cm and width $(x + 5)$ cm. Its perimeter is 40 cm. Find $x$. | $x = 4$ |
| 31 | Expand and simplify: $3(2x - 4) - 5x$. | $x - 12$ |
| 32 | If $n = 2^2 \times 5$, find the value of $3n - 7$. | 53 |
| 33 | I think of a number. I double it, then subtract 7. The result is -3. Find the number. | $n = 2$ |
| 34 | Three consecutive integers have a sum of 48. Find the three integers and write an equation to show your method. | 15, 16, 17 |
| 35 | Solve: $\dfrac{3x + 6}{2} = 9$. | $x = 4$ |
| 36 | The expression $5n - 3$ is evaluated for two consecutive negative integers. Show that the difference between the two results is always 5, whatever negative integers you choose. | The difference is always 5. |
| 37 | A mobile phone plan charges a fixed monthly fee of $\pounds 12$ plus $\pounds 3$ per GB of data used. Write a formula for the monthly cost $C$ in terms of $g$ (GB used). Use it to find how many GB were used if the bill was $\pounds 33$. | $C = 12 + 3g$; $g = 7$ GB |
| 38 | Alice thinks of a prime number $p$. She computes $p^2 - p$. Show algebraically that the result is always even. Verify with $p = 7$. | $p^2 - p = p(p-1)$ — always even. |
| 39 | Two numbers, $x$ and $y$, satisfy: $x + y = 14$ and $x - y = 4$. Find $x$ and $y$. | $x = 9,\ y = 5$ |
| 40 | The perimeter of a triangle is $37$ cm. One side is $(2x + 3)$ cm, a second side is $(3x - 1)$ cm, and the third side is $10$ cm. Find $x$ and the length of each side. | $x = 5$; sides: 13 cm, 14 cm, 10 cm |
| 41 | Explain why $5(x + 3)$ and $5x + 15$ are equal. What must 15 equal? Now expand $4(x - 2)$ and simplify $4(x - 2) + 6x$. | $c = 15$; $4(x-2) + 6x = 10x - 8$ |
| 42 | A number machine applies the rule "$\times 3$, then $- 8$". An input gives an output of $7$. Find the input. Then find the input that gives an output equal to the original input. | Input for output 7: $x = 5$. Fixed point: $x = 4$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | If $a = 6$, find a value of $b$ so that $3a + b = 21$. Justify your answer. | $b = 3$; because $3 \times 6 = 18$ and $18 + 3 = 21$ ✓ |
| 2 | Which of these is the correct algebraic convention: $n \times n$ or $n^2$? Explain why the other form is wrong. | $n^2$ is correct; repeated multiplication of the same variable is written using index notation |
| 3 | Write $$11x$ in two different ways as a sum of like terms. Show each is equivalent to the original. | E.g. $7x + 4x$ and $6x + 5x$; both simplify to $11x$ ✓ |
| 4 | Make up an equation whose solution is $x = 7$. Solve it to check. | E.g. $3x = 21$; solving: $x = 21 ÷ 3 = 7$ ✓ |
| 5 | A number of chairs $n$ is arranged in 5 equal rows. Write an equation for this and solve it to find $n$. | $5n = 45$... $n = 9$ chairs per row; equation from context: "total = rows × per row" |
| 6 | Expand $5(x + 2)$ and show that your answer gives the same value as the original expression when $x = 3$. | $5x + 10$; when $x = 3$: bracket gives $5(5) = 25$; expanded gives $15 + 10 = 25$ ✓ |
| 7 | Solve $3x + 4 = 19$ and check your answer by substituting back. | $x = 5$; check: $3(5) + 4 = 19$ ✓ |
| 8 | Write two different expressions that both describe "a number $n$ multiplied by 6, then subtract 5". Explain why they are the same. | $6n - 5$ and $6 \times n - 5$ are the same; $6n$ is the conventional shorthand |
| 9 | Decide: are $8y$ and $3y$ like terms? Simplify $8y - 3y$ and justify your step. | Yes, both have $y$; $8y - 3y = 5y$ because $8 - 3 = 5$ |
| 10 | In the expression $4x + 9$, identify the coefficient of $x$ and the constant. Explain what each part tells you about the expression. | Coefficient of $x$ is 4 (multiplies $x$); constant is 9 (never changes) |
| 11 | Little puzzle: what number $a$ makes $a \times 7 = 0$? How did you know? | $a = 0$. Multiplying anything by $0$ gives $0$, and $7$ on its own is not zero, so $a$ has to be the $0$. |
| 12 | Find the value of $2a + 3b$ when $a = 5$ and $b = 2$. | 16 |
| 13 | Simplify: $7x + 4y + 3x - 2y$. | $10x + 2y$ |
| 14 | Expand and simplify: $3(x + 5) + 7$. | $3x + 22$ |
| 15 | Solve: $5x - 2 = 23$. | $x = 5$ |
| 16 | Tickets to a show cost $\pounds {p}$ each. 7 friends buy tickets. Write an expression for the total cost, then find the cost when $p = 6$. | Expression: $7p$; cost = £42 |
| 17 | Find the value of $2x^2$ when $x = 5$. | 50 |
| 18 | Simplify: $8a + 7b + 1a - 4b$. | $9a + 3b$ |
| 19 | A rectangle has length $(x + 5)$ cm and width $6$ cm. Write a simplified expression for its perimeter. | $(2x + 22)$ cm |
| 20 | Expand the bracket: $3(4x - 7)$. | $12x - 21$ |
| 21 | Solve: $\dfrac{x}{4} + 3 = 8$. | $x = 20$ |
| 22 | Another little puzzle: what number $a$ makes $5(2 - a) = 0$? Try to spot the answer without doing lots of working. | $a = 2$. The $5$ isn't zero, so the bracket has to be zero — meaning $2 - a = 0$, so $a = 2$. |
| 23 | Simplify: $6x + 4x - 3x$. | $7x$ |
| 24 | Find the value of $3a^2 + 2a$ when $a = -3$. | 21 |
| 25 | Expand and simplify: $4(x + 3) + 3(x + 2)$. | $7x + 18$ |
| 26 | Solve: $5x + 9 = -1$. | $x = -2$ |
| 27 | A bag of apples costs ${p}p$ pence and a bag of oranges costs ${q}p$ pence. Write and simplify an expression for the total cost of 5 bags of apples and 2 bags of oranges. | $5p + 2q$ pence |
| 28 | Work out $4a - 3b$ when $a = 2$ and $b = -5$. | 23 |
| 29 | Solve: $6x + 2 = 2x + 14$. | $x = 3$ |
| 30 | A rectangle has length $(2x + 2)$ cm and width $(x + 4)$ cm. Its perimeter is 36 cm. Find $x$. | $x = 4$ |
| 31 | Expand and simplify: $4(3x - 2) - 6x$. | $6x - 8$ |
| 32 | If $n = 2^2 \times 3$, find the value of $3n - 4$. | 32 |
| 33 | I think of a number. I double it, then subtract 5. The result is -9. Find the number. | $n = -2$ |
| 34 | Three consecutive integers have a sum of 63. Find the three integers and write an equation to show your method. | 20, 21, 22 |
| 35 | Solve: $\dfrac{2x + 4}{3} = 6$. | $x = 7$ |
| 36 | The expression $5n - 3$ is evaluated for two consecutive negative integers. Show that the difference between the two results is always 5, whatever negative integers you choose. | The difference is always 5. |
| 37 | A mobile phone plan charges a fixed monthly fee of $\pounds 15$ plus $\pounds 4$ per GB of data used. Write a formula for the monthly cost $C$ in terms of $g$ (GB used). Use it to find how many GB were used if the bill was $\pounds 51$. | $C = 15 + 4g$; $g = 9$ GB |
| 38 | Alice thinks of a prime number $p$. She computes $p^2 - p$. Show algebraically that the result is always even. Verify with $p = 7$. | $p^2 - p = p(p-1)$ — always even. |
| 39 | Two numbers, $x$ and $y$, satisfy: $x + y = 20$ and $x - y = 6$. Find $x$ and $y$. | $x = 13,\ y = 7$ |
| 40 | The perimeter of a triangle is $46$ cm. One side is $(3x + 2)$ cm, a second side is $(2x - 3)$ cm, and the third side is $7$ cm. Find $x$ and the length of each side. | $x = 8$; sides: 26 cm, 13 cm, 7 cm |
| 41 | Explain why $6(x + 4)$ and $6x + 24$ are equal. What must 24 equal? Now expand $3(x - 5)$ and simplify $3(x - 5) + 7x$. | $c = 24$; $3(x-5) + 7x = 10x - 15$ |
| 42 | A number machine applies the rule "$\times 4$, then $- 10$". An input gives an output of $6$. Find the input. Then find the input that gives an output equal to the original input. | Input for output 6: $x = 4$. Fixed point: $x = \frac{10}{3}$ |
Problèmes — Solutions détaillées
**Think-of-a-number chain.** I think of a number. I triple it, then add 5. Then I subtract twice the original number. The result is 11. What is the number? Show your working using algebra.
6
**Perimeter puzzle.** A rectangle has length $(3x + 1)$ cm and width $(x + 3)$ cm. Its perimeter equals 48 cm. Find $x$, then calculate the area of the rectangle.
$x = 5$; area = 128 cm²
**Age problem.** Priya is $x$ years old. Her brother is 5 years older. Their mother is three times Priya's age. The sum of all three ages is 60. How old is each person?
Priya: 11, brother: 16, mother: 33
**Consecutive integer magic.** Alex claims: "Pick any three consecutive integers. Multiply the outer two together, then subtract the square of the middle one. You always get −1." (a) Test this with the consecutive integers 5, 6, 7. (b) Test it with −3, −2, −1. (c) Prove algebraically that it always works. Let the middle integer be $n$.
(a) −1 (b) −1 (c) Proof: $(n-1)(n+1) - n^2 = n^2 - 1 - n^2 = -1$, always.
**Equation from context.** Three friends share the cost of a meal equally. A service charge of £6 is added to the total before splitting. Each person ends up paying £14. Write and solve an equation to find the cost of the meal before the service charge.
£36
**Directed numbers + algebra.** I think of a number. I subtract 8 from it, then multiply the result by 3. The answer is $-6$. Form and solve an equation to find the number.
6
**The number machine — fixed points.** A number machine applies the rule: "Multiply by 3, then subtract 10." (a) What output does an input of 7 produce? (b) What input gives an output of −1? (c) Find the "fixed point" — the input that gives the same value as the output. Form and solve an equation. (d) What happens to inputs above the fixed point after many repeated applications? What about inputs below it?
(a) 11 (b) 3 (c) $x = 5$ (d) Inputs above 5 grow without bound; inputs below 5 decrease without bound.
**Temperature puzzle.** The temperature at midnight is $t$ °C. By 6 am it has risen by 7 °C. By noon it has doubled from the 6 am temperature. The noon temperature is 8 °C. Find $t$, the midnight temperature.
$t = -3$ °C
**Pattern and algebra.** A sequence of patterns is made from square tiles: - Pattern 1: 5 tiles - Pattern 2: 9 tiles - Pattern 3: 13 tiles (a) Write an expression for the number of tiles in pattern $n$. (b) Which pattern uses exactly 41 tiles? (c) Is there a pattern that uses exactly 100 tiles? Explain.
(a) $4n + 1$; (b) pattern 10; (c) No
**Staircase numbers.** A "staircase number" is any positive integer that can be written as the sum of two or more consecutive positive integers. For example, $9 = 4 + 5 = 2 + 3 + 4$. (a) Show that 15 is a staircase number in at least **three** different ways. (b) Test whether 16 is a staircase number. (Try all possible staircases with 2, 3, 4, or 5 consecutive integers.) (c) Show algebraically that the sum of $k$ consecutive integers starting from $n$ is $kn + \dfrac{k(k-1)}{2}$. (d) Which powers of 2 are staircase numbers? Make a conjecture.
(a) Three ways for 15: see working (b) 16 is not a staircase number (c) See algebraic derivation (d) No power of 2 is a staircase number.
**Balance puzzle.** On one side of a balance: 3 identical boxes and a 5 kg weight. On the other side: 7 identical boxes. All boxes are the same mass. The scales are balanced. Find the mass of one box. What total mass is on each side?
Each box: 1.25 kg; total each side: 8.75 kg
**Extended reasoning.** Jamie says: "If I square any integer and subtract the integer, the result is always even." (a) Test Jamie's claim for $n = -3$. (b) Prove algebraically that $n^2 - n$ is always even for any integer $n$. (c) If $n^2 - n = 30$, find all integer solutions for $n$.
(a) 12 (even); (b) proof below; (c) $n = 6$ and $n = -5$