Answer Key
11.10 Systems of Equations
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Solve $\;y = 2x + 1, \quad y = 7\;$ for $(x, y)$. | $x = 3$, $y = 7$ |
| 2 | Solve $\;x + y = 10, \quad x - y = 4\;$. | $x = 7$, $y = 3$ |
| 3 | Is $(x, y) = (4, 1)$ a solution of $\;3x + 2y = 14\;$? | Yes (LHS = 14) |
| 4 | Two straight lines meet at the point $(3, 5)$ on a graph. State the solution of the corresponding system of equations. | $x = 3$, $y = 5$ |
| 5 | Solve $\;\dfrac{x}{2} + y = 5, \quad y = 2\;$. | $x = 6$, $y = 2$ |
| 6 | Solve $\;y = 2x + 1, \quad 3x + y = 16\;$. | $x = 3$, $y = 7$ |
| 7 | A coffee costs $c$ and a tea costs $t$ francs. 2 coffees and 3 teas cost 15 francs. Write an equation in $c$ and $t$. | $2c + 3t = 15$ |
| 8 | Two equations $2x + 3y = 12$ and $4x + 6y = 24$ are graphed. State the number of solutions and explain. | Infinitely many solutions — the equations describe the same line. |
| 9 | Two equations $2x + 3y = 12$ and $2x + 3y = 18$ are graphed. State the number of solutions and explain. | No solutions — the lines are parallel (same slope, different intercept). |
| 10 | For $\;y = x^2, \; y = x + 6\;$, what type of system is this? | Linear–quadratic |
| 11 | Solve $\;2x + 3y = 12, \quad x - y = 1\;$. | $x = 3$, $y = 2$ |
| 12 | A coffee costs $x$ francs and a tea costs $y$ francs. 3 coffees + 2 teas = 19; 2 coffees + 4 teas = 22. Find $x$ and $y$. | $x = 4$, $y = 3.5$ |
| 13 | Solve $\;y = 3x - 4, \quad 2x + y = 11\;$. | $x = 3$, $y = 5$ |
| 14 | A taxi charges a fixed fee plus a rate per km. A 5 km ride costs CHF 12 and an 8 km ride costs CHF 18. Find the fixed fee and the rate per km. | Fixed CHF 2; rate CHF 2/km |
| 15 | The system $\;x + 2y = 7, \; 3x + ky = 5\;$ has solution $x = 1$. Find $k$ and $y$. | $y = 3$, $k = \dfrac{2}{3}$ |
| 16 | Solve $\;y = x^2, \; y = x + 6\;$. | $(3, 9)$ or $(-2, 4)$ |
| 17 | The graphs of $y = x + 1$ and $y = -x + 5$ meet at a single point. State the point and verify. | $(2, 3)$ |
| 18 | Use your GDC to solve $\;2.4x + 3.7y = 10, \quad 1.5x - 0.9y = 4.5\;$. Give $x$ and $y$ to 3 s.f. | $x \approx 2.84$, $y \approx 0.857$ |
| 19 | Solve $\;\dfrac{x}{2} + \dfrac{y}{3} = 5, \quad x + y = 12\;$. | $x = 6$, $y = 6$ |
| 20 | A graph shows demand $D = -2p + 100$ and supply $S = 3p + 5$ where $p$ is price. Find the equilibrium price and quantity. | Equilibrium price $p = 19$; quantity $= 62$ |
| 21 | Solve $\;x + y + z = 6, \quad 2x - y + z = 3, \quad x + 2y - z = 2\;$. | $x = 1$, $y = 2$, $z = 3$ |
| 22 | For what $k$ does $\;2x + 3y = 12, \; 4x + 6y = k\;$ have (a) inf many solutions, (b) no solution? | (a) $k = 24$; (b) $k \neq 24$ |
| 23 | Find the values of $k$ for which $\;y = kx, \; y = x^2 + 1\;$ has no real solutions. | $-2 < k < 2$ |
| 24 | A boat travels 12 km downstream and 12 km back upstream in 5 hours. Still-water speed $v$ km/h; current 2 km/h. Set up and solve. | $v \approx 5.52$ km/h |
| 25 | A business has cost $C = 200 + 8n$ and revenue $R = 12n$ for $n$ items. Find the break-even quantity. | $n = 50$ |
| 26 | Use technology to solve $\;x^2 + y^2 = 25, \; y = x + 1\;$. Give answers to 3 s.f. | $(3.42, 4.42)$ or $(-4.42, -3.42)$ |
| 27 | Find the value of $k$ for which $\;y = 2x + 1, \; y = kx - 3\;$ has no solution. | $k = 2$ |
| 28 | A person splits CHF 5000 between two investments: account A pays 3% per year and B pays 5%. The total annual interest is CHF 210. How much was placed in each? | A: CHF 2000; B: CHF 3000 |
| 29 | A quadratic $y = ax^2 + bx + c$ passes through $(1, 4)$, $(2, 9)$, $(3, 18)$. Use a 3×3 system to find $a$, $b$, $c$. | $a = 2$, $b = -1$, $c = 3$ |
| 30 | Find the intersections of the curves $y = x^2 - 4$ and $y = -x^2 + 2x$. | $(-1, -3)$ or $(2, 0)$ |
| 31 | Find all real solutions of $\;y = 2x + 1, \; y = x^2 - 2\;$. | $(-1, -1)$ or $(3, 7)$ |
| 32 | A quadratic $f(x) = ax^2 + bx + c$ satisfies $f(0) = 3$, $f(1) = 6$, $f(2) = 13$. Find $a$, $b$, $c$ and $f(-1)$. | $a = 2$, $b = 1$, $c = 3$; $f(-1) = 4$ |
| 33 | A theatre sells adult tickets for $a$ francs and children for $c$ francs. Monday: 80 adult + 30 child = CHF 1500. Tuesday: 60 adult + 50 child = CHF 1400. Find $a$ and $c$. | $a = 15$, $c = 10$ |
| 34 | Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \quad \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$. | $x = 2$, $y = 3$ |
| 35 | Quadratic regression on data through $(1, 5)$, $(2, 12)$, $(3, 23)$ gives $y = ax^2 + bx + c$. Set up and solve the system for $a, b, c$. | $a = 2$, $b = 1$, $c = 2$ |
| 36 | Two pipes fill a swimming pool. Pipe A alone takes 8 h; pipe B alone takes 12 h. How long do they take together? | $\dfrac{24}{5} = 4.8$ h (4 h 48 min) |
| 37 | Find the intersections of $\;xy = 6, \; y = x + 1\;$. | $(2, 3)$ or $(-3, -2)$ |
| 38 | Five years ago, Anna was 3 times as old as Ben. In 5 years, she will be twice as old. Find their current ages. | Anna 35, Ben 15 |
| 39 | For what value(s) of $k$ does the system $\;x + y + z = 3, \; x - y + 2z = k, \; 2x + 3y - z = 5\;$ have a unique solution? | Unique solution for all real $k$ (coefficient determinant $\neq 0$) |
| 40 | A printer charges either Plan A (CHF 30 fixed + CHF 8 per page) or Plan B (CHF 50 fixed + CHF 6 per page). (a) Find the number of pages for which both plans cost the same. (b) Which is cheaper for 25 pages? | (a) $n = 10$; (b) Plan B (CHF 200 < 230) |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Solve $\;y = 3x + -2, \quad y = 10\;$ for $(x, y)$. | $x = 4$, $y = 10$ |
| 2 | Solve $\;x + y = 12, \quad x - y = 2\;$. | $x = 7$, $y = 5$ |
| 3 | Is $(x, y) = (2, 5)$ a solution of $\;3x + 2y = 16\;$? | Yes (LHS = 16) |
| 4 | Two straight lines meet at the point $(3, 5)$ on a graph. State the solution of the corresponding system of equations. | $x = -2$, $y = 4$ |
| 5 | Solve $\;\dfrac{x}{2} + y = 8, \quad y = 3\;$. | $x = 10$, $y = 3$ |
| 6 | Solve $\;y = 2x + 1, \quad 3x + y = 21\;$. | $x = 4$, $y = 9$ |
| 7 | A coffee costs $c$ and a tea costs $t$ francs. 2 coffees and 3 teas cost 15 francs. Write an equation in $c$ and $t$. | $3c + 4t = 25$ |
| 8 | Two equations $2x + 3y = 12$ and $4x + 6y = 24$ are graphed. State the number of solutions and explain. | Infinitely many solutions. |
| 9 | Two equations $2x + 3y = 12$ and $2x + 3y = 18$ are graphed. State the number of solutions and explain. | No solutions — parallel lines. |
| 10 | For $\;y = x^2, \; y = x + 6\;$, what type of system is this? | Linear–quadratic |
| 11 | Solve $\;2x + 3y = 17, \quad x - y = 1\;$. | $x = 4$, $y = 3$ |
| 12 | A coffee costs $x$ francs and a tea costs $y$ francs. 3 coffees + 2 teas = 19; 2 coffees + 4 teas = 22. Find $x$ and $y$. | $x = 3$, $y = 3$ |
| 13 | Solve $\;y = 3x - 4, \quad 2x + y = 11\;$. | $x = -7$, $y = -11$ |
| 14 | A taxi charges a fixed fee plus a rate per km. A 5 km ride costs CHF 12 and an 8 km ride costs CHF 18. Find the fixed fee and the rate per km. | Fixed CHF 2; rate CHF 2/km |
| 15 | The system $\;x + 2y = 7, \; 3x + ky = 5\;$ has solution $x = 1$. Find $k$ and $y$. | $y = 5$, $k = 1$ |
| 16 | Solve $\;y = x^2, \; y = x + 6\;$. | $(1, 1)$ or $(-5, 25)$ |
| 17 | The graphs of $y = x + 1$ and $y = -x + 5$ meet at a single point. State the point and verify. | $(3, 5)$ |
| 18 | Use your GDC to solve $\;2.4x + 3.7y = 10, \quad 1.5x - 0.9y = 4.5\;$. Give $x$ and $y$ to 3 s.f. | $x \approx 2.31$, $y \approx 0.92$ |
| 19 | Solve $\;\dfrac{x}{2} + \dfrac{y}{3} = 5, \quad x + y = 12\;$. | $x = 3$, $y = 6$ |
| 20 | A graph shows demand $D = -2p + 100$ and supply $S = 3p + 5$ where $p$ is price. Find the equilibrium price and quantity. | Equilibrium price $p = 22$; quantity $= 54$ |
| 21 | Solve $\;x + y + z = 6, \quad 2x - y + z = 3, \quad x + 2y - z = 2\;$. | $x = 1$, $y = 3$, $z = 5$ |
| 22 | For what $k$ does $\;2x + 3y = 12, \; 4x + 6y = k\;$ have (a) inf many solutions, (b) no solution? | (a) $k = 10$; (b) $k \neq 10$ |
| 23 | Find the values of $k$ for which $\;y = kx, \; y = x^2 + 1\;$ has no real solutions. | $-2\sqrt{2} < k < 2\sqrt{2}$ |
| 24 | A boat travels 12 km downstream and 12 km back upstream in 5 hours. Still-water speed $v$ km/h; current 2 km/h. Set up and solve. | $v \approx 8.29$ km/h |
| 25 | A business has cost $C = 200 + 8n$ and revenue $R = 12n$ for $n$ items. Find the break-even quantity. | $n = 75$ |
| 26 | Use technology to solve $\;x^2 + y^2 = 25, \; y = x + 1\;$. Give answers to 3 s.f. | $(\pm 1.79, \pm 3.58)$ |
| 27 | Find the value of $k$ for which $\;y = 2x + 1, \; y = kx - 3\;$ has no solution. | No value — different intercepts mean lines are not the same when $k = 3$ either. |
| 28 | A person splits CHF 5000 between two investments: account A pays 3% per year and B pays 5%. The total annual interest is CHF 210. How much was placed in each? | A: CHF 5000; B: CHF 3000 |
| 29 | A quadratic $y = ax^2 + bx + c$ passes through $(1, 4)$, $(2, 9)$, $(3, 18)$. Use a 3×3 system to find $a$, $b$, $c$. | $a = 3$, $b = 0$, $c = 1$ |
| 30 | Find the intersections of the curves $y = x^2 - 4$ and $y = -x^2 + 2x$. | $(\pm 2, 5)$ |
| 31 | Find all real solutions of $\;y = 2x + 1, \; y = x^2 - 2\;$. | $(-1, 2)$ or $(4, 7)$ |
| 32 | A quadratic $f(x) = ax^2 + bx + c$ satisfies $f(0) = 3$, $f(1) = 6$, $f(2) = 13$. Find $a$, $b$, $c$ and $f(-1)$. | $a = 3$, $b = -2$, $c = 5$; $f(-1) = 10$ |
| 33 | A theatre sells adult tickets for $a$ francs and children for $c$ francs. Monday: 80 adult + 30 child = CHF 1500. Tuesday: 60 adult + 50 child = CHF 1400. Find $a$ and $c$. | $a = 12$, $c = 12.50$ |
| 34 | Solve $\;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \quad \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;$. | $x = 3$, $y = 4$ |
| 35 | Quadratic regression on data through $(1, 5)$, $(2, 12)$, $(3, 23)$ gives $y = ax^2 + bx + c$. Set up and solve the system for $a, b, c$. | $a = 1.5$, $b = 1$, $c = 1$ |
| 36 | Two pipes fill a swimming pool. Pipe A alone takes 8 h; pipe B alone takes 12 h. How long do they take together? | $\dfrac{18}{5} = 3.6$ h (3 h 36 min) |
| 37 | Find the intersections of $\;xy = 6, \; y = x + 1\;$. | $(3, 5)$ or $\left(-2, -5\right)$ — actually only $(3, 5)$ and $\left(\frac{-3}{1}, \ldots\right)$ — solve. |
| 38 | Five years ago, Anna was 3 times as old as Ben. In 5 years, she will be twice as old. Find their current ages. | Anna 22, Ben 7 |
| 39 | For what value(s) of $k$ does the system $\;x + y + z = 3, \; x - y + 2z = k, \; 2x + 3y - z = 5\;$ have a unique solution? | Unique solution for all real $k$ |
| 40 | A printer charges either Plan A (CHF 30 fixed + CHF 8 per page) or Plan B (CHF 50 fixed + CHF 6 per page). (a) Find the number of pages for which both plans cost the same. (b) Which is cheaper for 25 pages? | (a) $n \approx 6.67$ (or 7 pages); (b) Plan A (95 < 109) |
Problems — Worked Solutions
**2×2 system — substitution.** Solve $\begin{cases} y = 2x + 1 \\ 3x + 2y = 16 \end{cases}$ (a) Solve by substitution. (b) Verify your answer by checking both equations.
(a) $x = 2$, $y = 5$. (b) $5 = 2(2) + 1$ ✓; $3(2) + 2(5) = 16$ ✓.
**2×2 system — elimination.** Solve $\begin{cases} 3x + 2y = 16 \\ 5x - 2y = 8 \end{cases}$ (a) Solve by elimination. (b) Could you have spotted the answer faster? Justify briefly.
(a) $x = 3$, $y = 7/2$. (b) Yes — the $y$-terms cancel when adding.
**Coffee shop modelling.** A coffee shop sells small drinks for $s$ CHF and large drinks for $\ell$ CHF. - Monday: 30 small + 20 large = CHF 175. - Tuesday: 40 small + 25 large = CHF 230. (a) Write the system. (b) Solve for $s$ and $\ell$. (c) Predict Wednesday revenue: 50 small + 30 large.
(a) $30s + 20\ell = 175$; $40s + 25\ell = 230$. (b) $s = 2.5$, $\ell = 5$. (c) CHF 275.
**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.
(a) See working. (b) Pen $\approx$ CHF 2.12, pencil $\approx$ CHF 1.74, ruler = CHF 1.60.
**Linear–quadratic system.** Solve $\begin{cases} y = x^2 - 2 \\ y = x + 4 \end{cases}$ (a) Set the equations equal. (b) Solve the resulting quadratic. (c) State the two intersection points.
(a) $x^2 - 2 = x + 4$. (b) $x = 3$ or $x = -2$. (c) $(3, 7)$ or $(-2, 2)$.
**System from a graph.** Two lines $L_1$ and $L_2$ are given. $L_1$ has gradient $-2$ and passes through $(0, 5)$. $L_2$ passes through $(0, 1)$ and $(4, 9)$. (a) Find the equations of $L_1$ and $L_2$. (b) Find the intersection point.
(a) $L_1: y = -2x + 5$; $L_2: y = 2x + 1$. (b) $(1, 3)$.
**Boat speed.** A boat takes 5 hours to travel 12 km downstream and back upstream. The current is 2 km/h. (a) Let $v$ be the boat's still-water speed. Write the equation modelling the total time. (b) Solve for $v$ to 3 s.f.
(a) $\dfrac{12}{v + 2} + \dfrac{12}{v - 2} = 5$. (b) $v \approx 5.52$ km/h.
**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?
(a) $C_A = 8n + 30$; $C_B = 6n + 50$. (b) $n = 10$. (c) Plan B (CHF 200 vs CHF 230).
**Fit a quadratic [EXT].** A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$, $(2, 9)$, $(4, 21)$. (a) Write three equations. (b) Solve for $a$, $b$, $c$. (c) Predict $y$ at $x = 6$.
(a) $c = 5$; $4a + 2b + 5 = 9$; $16a + 4b + 5 = 21$. (b) $a = 1$, $b = 0$, $c = 5$. (c) $y(6) = 41$.
**Demand–supply.** In a market, demand $D(p) = -3p + 120$ and supply $S(p) = 2p + 10$, where $p$ 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.
(a) Price 22, quantity 54. (b) Price 23, quantity 51.
**No solution / infinitely many.** Consider $\;2x + 3y = 12, \; 4x + 6y = k$. (a) Find the value of $k$ for which the system has infinitely many solutions. (b) Find the values of $k$ for which the system has no solution. (c) For $k = 30$, sketch both lines and explain what you see.
(a) $k = 24$. (b) $k \neq 24$. (c) Parallel lines, never meet.
**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 $T$, the time to fill the empty tank when all three are open. (c) Solve for $T$ in hours.
(a) A: $\frac{1}{4}$/h; B: $\frac{1}{6}$/h; D: $-\frac{1}{8}$/h. (b) $\frac{1}{4} + \frac{1}{6} - \frac{1}{8} = \frac{1}{T}$. (c) $T \approx 3.43$ h (or $\frac{24}{7}$ h).