Problem-solving
8.8 Equations
Show all working. Partial marks are given for method.
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1**Linear-equation grand tour.** Solve each, showing your steps: (a) $4x + 7 = 31$ (b) $3(x - 4) = 18$ (c) $5x - 3 = 2x + 18$ (d) $\dfrac{x + 3}{4} = 7$
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2**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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3**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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4**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.
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5**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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6**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?
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7**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?
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8**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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9**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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10**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.
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11**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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12**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.
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