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Algebra, STD1 EQ-Bank 24

Zara is looking for a new mobile phone plan. She has found two plans that suit her needs and wants to work out which plan is cheaper depending on how many minutes she uses.

Plan \(\text{A}\):   $20 per month fixed charge plus $0.10 per minute

Plan \(\text{B}\):  $0.30 per minute, no fixed charge

Let  \(C\) = total monthly cost in dollars, and  \(m\) = number of minutes used.

  1. Plan \(\text{B}\) can be modelled by the equation  \(C=0.30m\).
  2. Write an equation for the monthly cost of Plan \(\text{A}\).   (1 mark)

  3. The graph of Plan \(\text{B}\) is provided on the grid below. Use the equation from (a) to add the graph of Plan \(\text{A}\) to the grid.   (2 marks)

      
  5. For how many minutes per month do both plans cost the same amount?   (1 mark)

  6. Zara uses an average of 140 minutes per month.
  7. Which plan she should choose and how much does she save compared to the other plan.   (1 mark)

Show Answers Only

a.    \(\text{Plan A: }C=20+0.10m\)

b.    

c.    \(100\ \text{minutes}\)

d.    \(\text{Plan A, cheaper by } \$8\)

Show Worked Solution

a.    \(\text{Plan A: }\ C=20+0.10m\)
  

b.    \(\text{Table of values}\)

\(\begin{array}{|c|c|c|c|c|c|} \hline m & 0 & 50 & 100 & 150 & 200 \\ \hline \text{Plan A} & 20 & 25 & 30 & 35 & 40 \\ \hline \end{array}\)
  


  

c.    \(\text{From the graph, the lines intersect at }\ m=100.\)

\(\therefore\ \text{Both plans cost the same at } 100\ \text{minutes}\)
  

d.    \(\text{Plan A:   }\ C=20+0.10\times140=\$34\)

\(\text{Plan B:   }\ C=0.30\times140=\$42\)

\(\text{Difference} = 42-34=\$8\)

\(\therefore\ \text{Zara should choose Plan A, which is \$8 cheaper than Plan B.}\)

Algebra, STD1 EQ-Bank 36

Two cyclists, Aiko and Ben, are riding along the same straight track in the same direction.

Aiko starts 2000 m ahead of Ben. Aiko rides at a constant speed of 250 metres/minute and Ben rides at a constant speed of 500 metres/minute.

Let    \(d\) = distance from the starting point in metres, and 

   \(t\) = time in minutes.

  1. The equation  \(d=2000+250t\)  models Aiko's distance from the starting point.
  2. Write an equation to model Ben's distance.   (1 mark)

  3. Use the equations from (a) to graph Aiko and Ben's journeys on the grid below.   (2 marks)

      
  4. After how many minutes does Ben catch Aiko?   (1 mark)

  5. How far has each cyclist travelled from their own starting point when they meet?   (2 marks)

Show Answers Only

a.    \(d=500t\)

b.    

c.    \(8\ \text{minutes}\)

d.    \(\text{Aiko travelled 2000 m, Ben travelled 4000 m.}\)

Show Worked Solution

a.    \(d=500t\)

b.    \(\text{Table of values:}\)

\(\begin{array}{|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline \text{Aiko} & 2000 & 2500 & 3000 & 3500 & 4000 & 4500 \\ \hline \text{Ben} & 0 & 1000 & 2000 & 3000 & 4000 & 5000 \\ \hline \end{array}\)
 

c.    \(\text{From the graph, the lines intersect at }\ t=8.\)

\(\therefore\ \text{Ben catches Aiko after 8 minutes}\)
 

d.    \(\text{When}\ \ t=8, d=4000:\)

\(\text{Aiko started 2000 m ahead, so the distance travelled}\)

\(=4000-2000=2000\ \text{m}\)

\(\text{Ben started at the origin, so the distance travelled =4000 m}\)

Algebra, STD1 EQ-Bank 34

Two water tanks sit side by side on a farm.

Tank A has a capacity of 1000 litres and is full. It is being emptied at a constant rate of 60 litres per minute.

At the same time, Tank B is empty and is being filled at a constant rate of 40 litres per minute.

Let    \(V\) = volume of water in litres, and 

   \(t\) = time in minutes.

  1. The equation  \(V=1000-60t\)  models the volume of water in Tank A.
  2. Write an equation to model the volume of water in Tank B.   (1 mark)

  3. Complete the table below and use the values to graph the volume of water in Tank A and Tank B on the grid below.   (3 marks)
     
          \(\begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{Tank A} & \ \ \ \ \ \ \ \  & 880 & 760 & 640 & 520 & 400 & 280 \\ \hline \text{Tank B} & 0 & \ \ \ \ \ \ \ \ \  & 160 & 240 & 320 & 400 & \ \ \ \ \ \ \ \  \\ \hline \end{array}\)
  
  2. After how many minutes do both tanks contain the same volume of water?   (1 mark)

  3. What is the volume of water in each tank at this time?   (1 mark)

Show Answers Only

a.    \(V=40t\)

b.    \(\text{Table of values:}\)

\(\begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{Tank A} & 1000 & 880 & 760 & 640 & 520 & 400 & 280 \\ \hline \text{Tank B} & 0 & 80 & 160 & 240 & 320 & 400 & 480 \\ \hline \end{array}\)

 

c.    \(10\ \text{minutes}\)

d.    \(400\ \text{litres}\)

Show Worked Solution

a.    \(V=40t\)

b.    \(\text{Table of values:}\)

\(\begin{array}{|c|c|c|c|c|c|c|c|} \hline t & 0 & 2 & 4 & 6 & 8 & 10 & 12 \\ \hline \text{Tank A} & 1000 & 880 & 760 & 640 & 520 & 400 & 280 \\ \hline \text{Tank B} & 0 & 80 & 160 & 240 & 320 & 400 & 480 \\ \hline \end{array}\)

 

c.    \(\text{From the graph, the lines intersect at }\ t=10.\)

\(\therefore\ \text{Both tanks contain the same volume after 10 minutes}\)
  

d.    \(\text{Method 1: Graphically}\)

\(\text{From the graph, when }\ t=10,\ \text{the volume in both tanks = 400 litres}\)
  

\(\text{Method 2: Algebraically}\)

\(\text{When }\ t=10:\)

\(\text{Tank A volume:}\ \ V=1000-60\times10=400\ \text{L}\)

\(\text{Tank B volume:}\ \ V=40\times10=400\ \text{L}\ \checkmark\)

\(\therefore\ \text{Each tank contains } 400\ \text{litres}\)

Algebra, STD1 A3 2021 HSC 29

In a park the only animals are goannas and emus. Let `x` be the number of goannas and let `y` be the number of emus.

The number of goannas plus the number of emus in the park is 31. Hence  `x + y = 31`.

Each goanna has four legs and each emu has two legs. In total the emus and goannas have 76 legs.

By writing another relevant equation and graphing both equations on the grid, find the number of goannas and the number of emus in the park.   (4 marks)

 

   
 

Number of goannas = _______________

Number of emus = _________________

Show Answers Only

`text{7 goannas, 24 emus}`

Show Worked Solution

`text{Total goanna legs} = 4x`

♦♦♦ Mean mark 8%.

`text{Total emu legs} = 2y`

`text{Total legs} = 76`

`4x + 2y` `= 76`  
`2x + y` `= 38\ \ …\ (1)`  
`x + y` `= 31\ \ …\ (2)`  

 

`text{Number of goannas}\ (x) = 7`

`text{Number of emus}\ (y) = 24`

Algebra, STD2 A4 EQ-Bank 5 MC

A computer application was used to draw the graphs of the equations

`x + y = 4`  and  `x-y = 4`

Part of the screen is shown.
 

Which row of the table correctly matches the equations with the lines drawn and identifies the solution when the equations are solved simultaneously?

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`A`

Show Worked Solution

`text(Line 1:) \ \ x + y = 4`

`text(Line 2:) \ \ x-y = 4`

`text(Intersection at) \ (4, 0).`

`=> A`

Algebra, STD2 A4 EQ-Bank 2 MC

A computer application was used to draw the graphs of the equations

`x-y = 4`  and  `x + y = 4`

Part of the screen is shown.

What is the solution when the equations are solved simultaneously?

  1. `x = 4, y = 4`
  2. `x = 4, y = 0`
  3. `x = 0, y = 4`
  4. `x = 0, y =-4`
Show Answers Only

`B`

Show Worked Solution

`text(Solution occurs at the intersection of the two lines.)`

`=> B`

Algebra, STD2 A4 2014 HSC 26d

Draw each graph on the grid below and hence solve the simultaneous equations.   (3 marks)

`y = 2x + 1`

`x-2y-4 = 0`
 

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`x=-2,\ y=-3`

Show Worked Solution

`text(Solution is at the intersection:)\ \ x=-2,\ y=-3`

Algebra, STD2 A4 EQ-Bank 19

The graph of the line  `y = 4-x`  is shown.
 


 

By graphing  `y = 2x + 1`  on the grid provided, find the point of intersection of  `y = 4-x`  and  `y = 2x + 1`.   (3 marks)

Show Answers Only

`(1, 3)`

Show Worked Solution

`text(Graphing)\ y = 2x + 1 :`

`ytext(-intercept = 1)`

`text(Gradient = 2)`
 

 
`:.\ text{Point of intersection is (1, 3).}`

Algebra, STD2 A4 EQ-Bank 18

A student was asked to solve the following simultaneous equations.

`y = 2x-8`

`x-4y + 3 = 0`

After graphing the equations, the student found the point of intersection to be `(5,2)`?

Is the student correct? Support your answer with calculations.   (2 marks)

Show Answers Only

`text(See Worked Solutions)`

Show Worked Solution

`text{Substitute (5, 2) into}\ \ y = 2x-8`

`text(LHS) = y = 2`

`text(RHS) = 2(5)-8 = 2=\ text{LHS}`
  

`text{Substitute (5, 2) into}\ \ x-4y + 3 = 0`

`text(LHS)= 5-4(2) + 3= 0=\ text(RHS)`

`=> (5,2)\ text(satisfies both equations.)`

`:.\ text(Student is correct.)`

Algebra, STD2 A4 EQ-Bank 17

`y` `= x + 5`
`y + 2x` `= 2`

 
Draw these two linear graphs on the number plane below and determine their intersection.   (3 marks)
 

 

 
Show Answers Only

`(-1,4)`

Show Worked Solution

`text(Table of coordinates:)\ \ y = x + 5`

\begin{array} {|c|c|c|}
\hline \quad x \quad & \ \  -5 \ \   & \ \  -4  \ \  & \ \ -2 \ \  & \quad 0  \quad\\
\hline y & 0 & 1 & 3 & 5 \\
\hline \end{array}

 
`text(Table of coordinates:)\ \ y + 2x = 2 \ => \ y =-2x + 2`

\begin{array} {|c|c|c|}
\hline \quad x \quad & \ \  -2 \ \   & \ \  -1  \ \  & \quad 0  \quad  & \quad 1  \quad\\
\hline y & 6 & 4 & 2 & 0 \\
\hline \end{array}
  

 
`text{From graph (and table), intersection occurs}`

`text(at)\ \ (-1,4).`

Algebra, STD2 A4 2017 HSC 17 MC

The graph of the line with equation  `y = 6-2x`  is shown.
 

When the graph of the line with equation  `y = x + 3`  is also drawn on this number plane, what will be the point of intersection of the two lines?

  1. `(0, 6)`
  2. `(1, 4)`
  3. `(2, 2)`
  4. `(3, 0)`
Show Answers Only

`B`

Show Worked Solution

`text(Method 1 – Graphically)`

`text(From graph, intersection at)\ (1,4)`
 

`=>B`
 

`text(Method 2 – Algebraically)`

`y` `= 6-2x` `…\ (1)`
`y` `= x + 3` `…\ (2)`

 
`text{Substitute (2) into (1)}`

`x + 3` `= 6-2x`
`3x` `= 3`
`x` `= 1`

 
`text(When)\ \ x = 1,\ y = 4`

`=>B`

Algebra, STD2 A4 2004 HSC 16 MC

George drew a correct diagram that gave the solution to the simultaneous equations

`y = 2x-5`  and  `y = x + 6`.

Which diagram did he draw?

Show Answers Only

`D`

Show Worked Solution

`text(By elimination)`

`y = 2x-5\ \ text(cuts the y-axis at)\ -5`

`:.\ text(Cannot be A or B)`

 

`y = x + 6\ \ text(cuts the y-axis at)\ 6`

`text(AND has a positive gradient)`

`:.\ text(Cannot be C)`

`=>  D`