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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.10 Systems of Equations

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
2×2 system — substitution. Solve

{y=2x+13x+2y=16\begin{cases} y = 2x + 1 \\ 3x + 2y = 16 \end{cases}

(a) Solve by substitution.
(b) Verify your answer by checking both equations.

Working space

2Problem 2 of 12
2×2 system — elimination. Solve

{3x+2y=165x−2y=8\begin{cases} 3x + 2y = 16 \\ 5x - 2y = 8 \end{cases}

(a) Solve by elimination.
(b) Could you have spotted the answer faster? Justify briefly.

Working space

3Problem 3 of 12
Coffee shop modelling. A coffee shop sells small drinks for ss CHF and large drinks for ℓ\ell CHF.

- Monday: 30 small + 20 large = CHF 175.
- Tuesday: 40 small + 25 large = CHF 230.

(a) Write the system.
(b) Solve for ss and ℓ\ell.
(c) Predict Wednesday revenue: 50 small + 30 large.

Working space

4Problem 4 of 12
Three-variable system [EXT]. A school orders pens, pencils and rulers.

- 5 pens + 3 pencils + 2 rulers = CHF 19
- 2 pens + 4 pencils + 3 rulers = CHF 16
- 1 pen + 2 pencils + 4 rulers = CHF 12

(a) Set up the system.
(b) Solve.

Working space

5Problem 5 of 12
Linear–quadratic system. Solve

{y=x2−2y=x+4\begin{cases} y = x^2 - 2 \\ y = x + 4 \end{cases}

(a) Set the equations equal.
(b) Solve the resulting quadratic.
(c) State the two intersection points.

Working space

6Problem 6 of 12
System from a graph. Two lines L1L_1 and L2L_2 are given. L1L_1 has gradient −2-2 and passes through (0,5)(0, 5). L2L_2 passes through (0,1)(0, 1) and (4,9)(4, 9).

(a) Find the equations of L1L_1 and L2L_2.
(b) Find the intersection point.

Working space

7Problem 7 of 12
Boat speed. A boat takes 5 hours to travel 12 km downstream and back upstream. The current is 2 km/h.

(a) Let vv be the boat's still-water speed. Write the equation modelling the total time.
(b) Solve for vv to 3 s.f.

Working space

8Problem 8 of 12
Break-even. A school trip is offered with two pricing plans.

- Plan A: CHF 30 fixed minibus + CHF 8 per student.
- Plan B: CHF 50 fixed minibus + CHF 6 per student.

(a) Write linear cost equations for each plan.
(b) Find the number of students at which both plans cost the same.
(c) Which plan is cheaper for 25 students?

Working space

9Problem 9 of 12
Fit a quadratic [EXT]. A quadratic y=ax2+bx+cy = ax^2 + bx + c passes through (0,5)(0, 5), (2,9)(2, 9), (4,21)(4, 21).

(a) Write three equations.
(b) Solve for aa, bb, cc.
(c) Predict yy at x=6x = 6.

Working space

10Problem 10 of 12
Demand–supply. In a market, demand D(p)=−3p+120D(p) = -3p + 120 and supply S(p)=2p+10S(p) = 2p + 10, where pp is price.

(a) Find the equilibrium price and quantity.
(b) If a tax of CHF 5 is added per unit (shifting supply up by 5), find the new equilibrium.

Working space

11Problem 11 of 12
No solution / infinitely many. Consider   2x+3y=12,  4x+6y=k\;2x + 3y = 12, \; 4x + 6y = k.

(a) Find the value of kk for which the system has infinitely many solutions.
(b) Find the values of kk for which the system has no solution.
(c) For k=30k = 30, sketch both lines and explain what you see.

Working space

12Problem 12 of 12
Modelling — rates of work [EXT]. A water tank can be filled by pipe A alone in 4 hours, by pipe B alone in 6 hours. A drain D empties the full tank in 8 hours.

(a) Express each rate (tank per hour) as a fraction.
(b) Set up an equation for TT, the time to fill the empty tank when all three are open.
(c) Solve for TT in hours.

Working space