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Algebra, MET2 2023 VCE SM-Bank 1 MC

\begin{aligned}
& x-2 y=3 \\
& 2 y-z=4
\end{aligned}

Which one of the following correctly describes the general solution to the system of linear equations given above?

  1. \(x=k, \quad y=\dfrac{1}{2}(k+3), \ z=k-1\), for all \(k \in R\)
  2. \(x=k, \quad y=\dfrac{1}{2}(k+3), \ z=k+1\), for all \(k \in R\)
  3. \(x=k, \quad y=\dfrac{1}{2}(k-3), \ z=k+7\), for all \(k \in R\)
  4. \(x=k, \quad y=\dfrac{1}{2}(k-3), \ z=k-7\), for all \(k \in R\)
  5. \(x=k, \quad y=\dfrac{1}{2}(k+3), \ z=k-7\), for all \(k \in R\)
Show Answers Only

\(D\)

Show Worked Solution

\(\text{When }x=k\quad (1)\)

\(2y\) \(=k-3\)
\(y\) \(=\dfrac{1}{2}(k-3)\quad (2)\)
\(\text{and}\ z\) \(=2y-4\)
\(z\) \(=2\times\dfrac{1}{2}(k-3)-4\)
  \(=k-7\quad (3)\)

  
\(\text{Using CAS:}\)

\(\Rightarrow D\)

Algebra, MET2 2007 VCAA 5 MC

The simultaneous linear equations

`mx + 12y = 24`

`3x + my = m`

have a unique solution only for

  1. `m = 6 or m = – 6`
  2. `m = 12 or m = 3`
  3. `m in R\ text(\){– 6, 6}`
  4. `m = 2 or m = 1`
  5. `m in R\ text(\){– 12, – 3}`
Show Answers Only

`C`

Show Worked Solution
`mx + 12y` `=24`
`y` `=-m/12 x +2\ \ …\ (1)`
`3x + my` `=m`
`y` `= -3/m x+1\ \ …\ (2)`

 

`text(Unique solution occurs when)\ \ m_1!= m_2,`

♦♦ Mean mark 36%.
`-m/12` `!=-3/m`
`m^2` `!= 36`
`:. m` `!= +- 6`

`=>   C`

Algebra, MET1 SM-Bank 25

Solve these simultaneous equations to find the values of  `x`  and  `y`.    (3 marks)

`y = 2x + 1`

`x − 2y − 4 = 0`

Show Answers Only

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

Show Worked Solution

`text(Solution 1 – Substitution)`

`y = 2x + 1\ \ \ \ \ …\ text{(i)}`

`x\ – 2y\ – 4 = 0\ \ \ \ \ …\ text{(ii)}`

 

`text(Substitute)\ \ y = 2x + 1\ \ text(into)\ text{(ii)}`

`x\ – 2(2x + 1)\ – 4` `= 0`
`x\ – 4x\ – 2\ – 4` `= 0`
`-3x\ – 6` `= 0`
`3x` `= -6`
`x` `= -2`

 
`text(Substitute)\ \ x = –2\ \ text(into)\ text{(i)}`

`y = 2(–2) + 1 = -3`
 

`:.\ text(Solution is)\ x = -2,\ y = -3`
 

`text(Alternative Solution – Elimination)`

`y = 2x + 1\ \ \ \ \ …\ text{(i)}`

`x\ – 2y\ – 4 = 0\ \ \ \ \ …\ text{(ii)}`
 

`text(Multiply)\ text{(i)} xx 2`

`2y` `= 4x + 2`
`-4x + 2y\ – 2` `= 0\ \ \ \ \ …\ text{(iii)}`

 

`text{Add  (ii) + (iii)}`

`-3x\ – 6` `= 0`
`x` `= -2`
`y` `= -3\ \ text{(see Solution 1)}`

Algebra, MET2 2009 VCAA 1 MC

The simultaneous linear equations

`kx - 3y = 0`

`5x - (k + 2)y = 0`

where `k` is a real constant, have a unique solution provided

  1. `k in {– 5, 3}`
  2. `k in R\ text(\){– 5, 3}`
  3. `k in {– 3, 5}`
  4. `k in R\ text(\){– 3, 5}`
  5. `k in R\ text(\){0}`
Show Answers Only

`B`

Show Worked Solution

`text(Unique solution occurs when:)`

♦ Mean mark 49%.
`m_1` `!= m_2`
`k/5` `!= (– 3)/(– (k + 2))`
`k^2+2k-15` `!=0`
`(k+5)(k-3)` `!=0`
`:. k` `!= – 5, 3`

`=>   B`

Algebra, MET1 2011 VCAA 6

Consider the simultaneous linear equations

`kx - 3y` `= k + 3`
`4x + (k + 7) y` `= 1`

where `k` is a real constant.

  1. Find the value of `k` for which there are infinitely many solutions.  (3 marks)
  2. Find the values of `k` for which there is a unique solution.  (1 mark)
Show Answers Only
  1. `– 4`
  2. `k in R\ text(\)\ {– 4, – 3}`
Show Worked Solution

a.   `text(Infinite solutions if gradients and)`

♦ Mean mark 39%.
MARKER’S COMMENT: A number of solutions are possible here, including using the determinant.

`y text(-intercepts are the same.)`

`kx-3y` `=k+3`
`3y` `=kx-k-3`
`y` `=k/3 x – (k+3)/3`
`:.m_1=k/3 and c_1= -((k+3)/3)`

 

`4x + (k + 7) y` `=1`
`y` `=((-4)/(k+7)) x + 1/(k+7)`
`:. m_2=(-4)/(k+7) and c_2=1/(k+7)`
   

 

`text(Equating gradients and)\ y text(-intercepts:)`

`m_1` `= m_2` `\ \ \ and` `\ \ \ c_1` `=c_2`
`k/3` `= (– 4)/(k + 7)`   `-((k+3)/3)` `= 1/(k+7)`
`k^2 + 7k` `= – 12`   `-(k+3)(k+7)` `=3`
`k^2 + 7k + 12` `= 0`   `k^2+10k+24` `=0`
`(k + 3) (k + 4)` `= 0`   `(k+6)(k+4)` `= 0`
`k` `= – 3, – 4`   `k`  `= – 6, – 4`

 

`:. k = – 4\ \ \ text{(satisfies both)}`

 

b.   `text(Unique solution if:)`

♦♦ Mean mark 33%.

`m_1 != m_2`

`:. k in R\ text(\)\ {– 4, – 3}`

Algebra, MET1 SM-Bank 28

Consider the simultaneous linear equations below.

 `4x - 2y = 18`

`3x + ky = 10`   (3 marks)

where  `k`  is a real constant.

  1. What are the values of  `k`  where no solutions exist?   (3 marks)
  2. What values of  `k`  do the simultaneous equations have a unique solution?   (1 mark)
Show Answers Only
  1. `k=-3/2`
  2. `k\ in R text(\) {-3/2}` 
Show Worked Solution
a.    `4x – 2y` `=18`
  `y` `=2x-9\ \ …\ (1)`

`=> m_1 = 2,\ \ c_1=-9`

 

`3x +ky` `=10`
`y` `=-3/k x +10/k\ \ …\ (2)`

`=> m_2 = – 3/k,\ \ c_2=10/k`

 

`text(No solution if)\ \ m_1=m_2,  and  c_1!=c_2.`

`-3/k` `=2`
`k` `=- 3/2`

`text(When)\ \ k=-3/2,  c_1!=c_2.`

`:.\ text(No solution when)\ \ k=-3/2.`

 

b.   `text(A unique solution exists when)\ \ m_1 != m_2,`

`k in R\ text(\) {-3/2}`