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Pack B — Fluency

Ecolint Campus des NationsMathematics
Year 11 · 11.10 Systems of Equations

Pack B · Fluency

Name: _________________________________
Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10
  1. 1.
    Solve   y=3x+−2,y=10  \;y = 3x + -2, \quad y = 10\; for (x,y)(x, y).
     
  2. 2.
    Solve   x+y=12,x−y=2  \;x + y = 12, \quad x - y = 2\;.
     
  3. 3.
    Is (x,y)=(2,5)(x, y) = (2, 5) a solution of   3x+2y=16  \;3x + 2y = 16\;?
     
  4. 4.
    Two straight lines meet at the point (3,5)(3, 5) on a graph. State the solution of the corresponding system of equations.
     
  5. 5.
    Solve   x2+y=8,y=3  \;\dfrac{x}{2} + y = 8, \quad y = 3\;.
     
  6. 6.
    Solve   y=2x+1,3x+y=21  \;y = 2x + 1, \quad 3x + y = 21\;.
     
  7. 7.
    A coffee costs cc and a tea costs tt francs. 2 coffees and 3 teas cost 15 francs. Write an equation in cc and tt.
     
  8. 8.
    Two equations 2x+3y=122x + 3y = 12 and 4x+6y=244x + 6y = 24 are graphed. State the number of solutions and explain.
     
  9. 9.
    Two equations 2x+3y=122x + 3y = 12 and 2x+3y=182x + 3y = 18 are graphed. State the number of solutions and explain.
     
  10. 10.
    For   y=x2,  y=x+6  \;y = x^2, \; y = x + 6\;, what type of system is this?
     
SilverQuestions 11–20
  1. 11.
    Solve   2x+3y=17,x−y=1  \;2x + 3y = 17, \quad x - y = 1\;.
     
  2. 12.
    A coffee costs xx francs and a tea costs yy francs. 3 coffees + 2 teas = 19; 2 coffees + 4 teas = 22. Find xx and yy.
     
  3. 13.
    Solve   y=3x−4,2x+y=11  \;y = 3x - 4, \quad 2x + y = 11\;.
     
  4. 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.
     
  5. 15.
    The system   x+2y=7,  3x+ky=5  \;x + 2y = 7, \; 3x + ky = 5\; has solution x=1x = 1. Find kk and yy.
     
  6. 16.
    Solve   y=x2,  y=x+6  \;y = x^2, \; y = x + 6\;.
     
  7. 17.
    The graphs of y=x+1y = x + 1 and y=−x+5y = -x + 5 meet at a single point. State the point and verify.
     
  8. 18.
    Use your GDC to solve   2.4x+3.7y=10,1.5x−0.9y=4.5  \;2.4x + 3.7y = 10, \quad 1.5x - 0.9y = 4.5\;. Give xx and yy to 3 s.f.
     
  9. 19.
    Solve   x2+y3=5,x+y=12  \;\dfrac{x}{2} + \dfrac{y}{3} = 5, \quad x + y = 12\;.
     
  10. 20.
    A graph shows demand D=−2p+100D = -2p + 100 and supply S=3p+5S = 3p + 5 where pp is price. Find the equilibrium price and quantity.
     
GoldQuestions 21–30
  1. 21.
    Solve   x+y+z=6,2x−y+z=3,x+2y−z=2  \;x + y + z = 6, \quad 2x - y + z = 3, \quad x + 2y - z = 2\;.
     
  2. 22.
    For what kk does   2x+3y=12,  4x+6y=k  \;2x + 3y = 12, \; 4x + 6y = k\; have (a) inf many solutions, (b) no solution?
     
  3. 23.
    Find the values of kk for which   y=kx,  y=x2+1  \;y = kx, \; y = x^2 + 1\; has no real solutions.
     
  4. 24.
    A boat travels 12 km downstream and 12 km back upstream in 5 hours. Still-water speed vv km/h; current 2 km/h. Set up and solve.
     
  5. 25.
    A business has cost C=200+8nC = 200 + 8n and revenue R=12nR = 12n for nn items. Find the break-even quantity.
     
  6. 26.
    Use technology to solve   x2+y2=25,  y=x+1  \;x^2 + y^2 = 25, \; y = x + 1\;. Give answers to 3 s.f.
     
  7. 27.
    Find the value of kk for which   y=2x+1,  y=kx−3  \;y = 2x + 1, \; y = kx - 3\; has no solution.
     
  8. 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?
     
  9. 29.
    A quadratic y=ax2+bx+cy = ax^2 + bx + c passes through (1,4)(1, 4), (2,9)(2, 9), (3,18)(3, 18). Use a 3×3 system to find aa, bb, cc.
     
  10. 30.
    Find the intersections of the curves y=x2−4y = x^2 - 4 and y=−x2+2xy = -x^2 + 2x.
     
PlatinumQuestions 31–40
  1. 31.
    Find all real solutions of   y=2x+1,  y=x2−2  \;y = 2x + 1, \; y = x^2 - 2\;.
     
  2. 32.
    A quadratic f(x)=ax2+bx+cf(x) = ax^2 + bx + c satisfies f(0)=3f(0) = 3, f(1)=6f(1) = 6, f(2)=13f(2) = 13. Find aa, bb, cc and f(−1)f(-1).
     
  3. 33.
    A theatre sells adult tickets for aa francs and children for cc francs. Monday: 80 adult + 30 child = CHF 1500. Tuesday: 60 adult + 50 child = CHF 1400. Find aa and cc.
     
  4. 34.
    Solve   1x+1y=56,1x−1y=16  \;\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{5}{6}, \quad \dfrac{1}{x} - \dfrac{1}{y} = \dfrac{1}{6}\;.
     
  5. 35.
    Quadratic regression on data through (1,5)(1, 5), (2,12)(2, 12), (3,23)(3, 23) gives y=ax2+bx+cy = ax^2 + bx + c. Set up and solve the system for a,b,ca, b, c.
     
  6. 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?
     
  7. 37.
    Find the intersections of   xy=6,  y=x+1  \;xy = 6, \; y = x + 1\;.
     
  8. 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.
     
  9. 39.
    For what value(s) of kk does the system   x+y+z=3,  x−y+2z=k,  2x+3y−z=5  \;x + y + z = 3, \; x - y + 2z = k, \; 2x + 3y - z = 5\; have a unique solution?
     
  10. 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?