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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 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 8%.

`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 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`

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`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`

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`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)`

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

`ytext(-intercept = 1)`

`text(Gradient = 2)`
 

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

Algebra, STD2 A4 SM-Bank 6

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)

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`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 = RHS)`
 

`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 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)
 

 

 
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`(−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`