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Functions, 2ADV EQ-Bank 24

Solve the following simultaneous equations

\(6 x-7 y=-6\)

\(4 x+2 y=16\)   (2 marks)

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\(x=\dfrac{5}{2}, \ y=3\)

Show Worked Solution

\(6x-7y=-6\ \ldots\ (1)\)

\(4x+2y=16\ \ldots\ (2)\)
 

\(\text{Mult \((1) \times 2\ \) and \(\ (2) \times 3\):}\)

\(12 x-14 y=-12\ \ldots\ \left(1^{\prime}\right)\)

\(12 x+6 y=48\ \ldots\ \left(2^{\prime}\right)\)
 

\(\text{Subtract}\ \ \left(2^{\prime}\right)-\left(1^{\prime}\right):\)

\(20 y=60 \ \Rightarrow \ y=3\)
 

\(\text{Substitute} \ \ y=3 \ \ \text{into (2):}\)

\(4 x+6=16 \ \Rightarrow \ x=\dfrac{5}{2}\)

Functions, 2ADV EQ-Bank 12

Solve the following simultaneous equations:

\(3 x-4 y=5\)

\(x+2 y=15\).   (2 marks)

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\(x=7, \ y=4\)

Show Worked Solution

\(3 x-4 y=5\ \ldots\ (1)\)

\( x+2 y=15\ \ldots\ (2)\)

\(\text{Multiply (2)}  \times 3:\)

\(3 x+6 y=45\ \ldots\ \left(2^{\prime}\right) \)
 

\(\text{Subtract} \ \left(2^{\prime}\right)-(1)\) :

\(10 y=40 \ \Rightarrow \ y=4\)
 

\(\text{Substitute} \ \ y=4 \ \ \text{into (2):}\)

\(x+2 \times 4=15 \ \Rightarrow x=7\)

Functions, 2ADV F1 2020 HSC 11

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

  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}`

`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 A2 2019 HSC 14 MC

Last Saturday, Luke had 165 followers on social media. Rhys had 537 followers. On average, Luke gains another 3 followers per day and Rhys loses 2 followers per day.

If `x` represents the number of days since last Saturday and `y` represents the number of followers, which pair of equations model this situation?

A.  `text(Luke:)\ \ y = 165x + 3`

 

`text(Rhys:)\ \ y = 537x-2`
B. `text(Luke:)\ \ y = 165 + 3x`

 

`text(Rhys:)\ \ y = 537-2x`
C. `text(Luke:)\ \ y = 3x + 165`

 

`text(Rhys:)\ \ y = 2x-537`
D. `text(Luke:)\ \ y = 3 + 165x`

 

`text(Rhys:)\ \ y = 2-537x`
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`B`

Show Worked Solution

`text(Luke starts with 165 and adds 3 per day:)`

`y = 165 + 3x`

`text(Rhys starts with 537 and loses 2 per day:)`

`y = 537-2x`

`=> B`

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`
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`B`

Show Worked Solution

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

`=> B`