Solving systems of equations graphicallyIB MYP Maths Standard: Subtopic test

1. 1
   Two straight lines are drawn on the same axes: $y = 2x + 1$ and $y = -x + 7$.

   (a)

   Which point is the solution of the simultaneous equations $y=2x+1$ and $y=-x+7$?

   [1 mark]

   • A$(5, 2)$
   • B$(1, 3)$
   • C$(2, 5)$
   • D$(3, 4)$

   (b)

   Which equation can be solved to find the $x$-coordinate of the point where the two lines meet?

   [1 mark]

   • A$2x+1=0$
   • B$2x+1=-x-7$
   • C$2x+1+(-x+7)=0$
   • D$2x+1=-x+7$

   (c)

   Use algebra to find the coordinates of the point where the two lines meet.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Tank A holds $50$ litres of water and is filled at $5$ litres per minute, so after $t$ minutes it holds $V = 50 + 5t$ litres. Tank B holds $20$ litres and is filled at $8$ litres per minute, so after $t$ minutes it holds $V = 20 + 8t$ litres.

   (a)

   What does the point where the two lines cross represent?

   [1 mark]

   • AThe time when both tanks are full.
   • BThe time when both tanks hold the same volume of water.
   • CThe volume of water in each tank at the start.
   • DThe time when both tanks are empty.

   (b)

   After $4$ minutes, which tank holds more water?

   [1 mark]

   • ATank A, by $18$ litres
   • BTank B, because it fills faster
   • CTank B, by $18$ litres
   • DNeither; they hold the same volume

   (c)

   Find the time at which the two tanks hold the same volume of water, and state that volume.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The graphs of $f(x) = x^2 - 2$ and $g(x) = x + 4$ are drawn on the same axes.

   (a)

   Show that the $x$-coordinates of the points where the two graphs meet satisfy $x^2-x-6=0$, and solve this equation.

   [3 marks]

   (b)

   Find the coordinates of both points where the graphs meet.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A school is choosing a company to hire a hall for a prom. The cost $C$ (in AED) for $g$ guests is: Company P, $C = 200 + 8g$; Company Q, $C = 16g$; Company R, $C = 12g + 100$.

   (a)

   (i) Find the number of guests for which Company P and Company Q charge the same, and the cost.
   (ii) The school has $40$ guests. Which of P and Q is cheaper?

   [6 marks]

   (b)

   A teacher says that all three companies charge the same amount for $25$ guests. Decide whether she is correct. Then find which company is cheapest for $20$ guests and which is cheapest for $40$ guests.

   [6 marks]

   Total for question 4: 12 marks

End of questions