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Algebra, STD2 A4 2021 HSC 34

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 on the following page, find the number of goannas and the number of emus in the park.  (4 marks)

   
 

Number of goannas = _______________

Number of emus = _________________

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`text{7 goannas, 24 emus}`

Show Worked Solution

`text{Total goanna legs} = 4x`

♦♦♦ Mean mark 29%.

`text{Total emu legs} = 2y`

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

 

`text{Number of goannas} = 7`

`text{Number of emus} = 24`

Algebra, STD2 A4 2020 HSC 24

There are two tanks on a property, Tank A and Tank B. Initially, Tank A holds 1000 litres of water and Tank B is empty.

  1. Tank A begins to lose water at a constant rate of 20 litres per minute.

     

    The volume of water in Tank A is modelled by  `V = 1000 - 20t`  where `V` is the volume in litres and  `t`  is the time in minutes from when the tank begins to lose water.

     

    On the grid below, draw the graph of this model and label it as Tank A.   (1 mark)
     

     

  2. Tank B remains empty until  `t=15`  when water is added to it at a constant rate of 30 litres per minute.
     By drawing a line on the grid (above), or otherwise, find the value of  `t`  when the two tanks contain the same volume of water.  (2 marks)

  3. Using the graphs drawn, or otherwise, find the value of  `t`  (where  `t > 0`) when the total volume of water in the two tanks is 1000 litres.  (1 mark)

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  1.  `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}` 
      
  2. `29 \ text{minutes}`
  3. `45 \ text{minutes}`
Show Worked Solution

a.     `text{T} text{ank} \ A \ text{will pass trough (0, 1000) and (50, 0)}` 
 

 

b.   `text{T} text{ank} \ B \ text{will pass through (15, 0) and (45, 900)}`  
 

   

`text{By inspection, the two graphs intersect at} \ \ t = 29 \ text{minutes}`

 
c.   `text{Strategy 1}`

♦♦ Mean mark part (c) 22%.

`text{By inspection of the graph, consider} \ \ t = 45`

`text{T} text{ank A} = 100 \ text{L} , \ text{T} text{ank B} =900 \ text{L} `

`:.\ text(Total volume = 1000 L when  t = 45)`
  

`text{Strategy 2}`

`text{Total Volume}` `=text{T} text{ank A} + text{T} text{ank B}`
`1000` `= 1000 – 20t + (t – 15) xx 30`
`1000` `= 1000 – 20t + 30t – 450 `
`10t` `= 450`
`t` `= 45 \ text{minutes}`

Algebra, STD2 A4 SM-Bank 7 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 SM-Bank 6 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`
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`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`   (3 marks)
 

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

Show Worked Solution

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

Algebra, STD2 A4 SM-Bank 7

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)

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`(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 SM-Bank 5

`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`

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

 
`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?

A.     `(0, 6)`

B.     `(1, 4)`

C.     `(2, 2)`

D.     `(3, 0)`

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`text(B)`

Show Worked Solution

`text(Solution 1)`

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

`=>B`
 

`text(Solution 2)`

`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?

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`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`