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Consider the system of simultaneous equations below containing the parameter \(k\).
| \(kx+5y\) | \(=k+5\) |
| \(4x+(k+1)y\) | \(=0\) |
The value(s) of \(k\) for which the system of equations has infinite solutions are
\(B\)
| \(kx+5y\) | \(=k+5\) | \(\ \ \rightarrow\ \ \ \ \) | \(y\) | \(=-\dfrac{k}{5}x+\dfrac{k+5}{5}\) |
| \(4x+(k+1)y\) | \(=0\) | \(\ \ \rightarrow\ \ \ \ \) | \(y\) | \(=-\dfrac{4}{k+1}x+0\) |
\(\text{Infinite solutions when gradients and }y\text{-intercepts equal.}\)
\(\text{Equating gradients (by CAS):}\)
\(-\dfrac{k}{5}=-\dfrac{4}{k+1}\ \ \Rightarrow\ \ k=4\ \text{ or}\ -5\)
\(\text{Equating intercepts:}\)
\(\dfrac{k+5}{5}=0\ \ \Rightarrow \ \ k=-5\)
\(k=-5\ \ \text{satisfies both equations (infinite solutions)}.\)
\(\Rightarrow B\)
Consider the system of equations
`kx-5y=4+k`
`3x+(k+8) y=-1`
Determine the value of `k` for which the system of equations above has an infinite number of solutions. (3 marks)
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`k=-3` for infinite solutions
Making `y` the subject of each equation:
| `k x-5 y` | `=4+k ` | |
| `y` | `=\frac{k}{5} x+\frac{(-4-k)}{5}` | (1) |
| `3x+(k+8) y` | `=-1` | |
| `y` | `=\frac{-3}{k+8} \times-\frac{1}{k+8}` | (2) |
From equations (1) and (2)
`m_1=\frac{k}{5}` and `m_2=-\frac{3}{k+8}`
An infinite number of solutions occur when `m_1=m_2`
| `\frac{k}{5}` | `=-\frac{3}{k+8}` | |
| `k(k+8)` | `=-15` | |
| `k^2+8 k+15` | `=0` | |
| `(k+3)(k+5)` | `=0` |
`\therefore k=-5,-3`
Let `c_1` and `c_2` be the `y`-intercepts of equations (1) and (2)
then `c_1 =\frac{-k-4}{5}` and `c_2 =-\frac{1}{k+8}`
`\therefore \frac{-k-4}{5}=-\frac{1}{k+8}`
| `(k+4)(k+8)` | `=5` | |
| `k^2+12 k+27` | `=0` | |
| `(k+3)(k+9)` | `=0` | |
| `k=-3 \text {, }` | `k=-9` |
`\therefore k=-3` for infinite solutions and only value to satisfy both equations.
The simultaneous linear equations `2y + (m - 1) x = 2` and `my + 3x = k` have infinitely many solutions for
`D`
| `2y + (m – 1)x` | `= 2\ \ =>\ \ y= -((m-1)/(2)) x + 1` |
| `my + 3x` | `= k\ \ =>\ \ y=-(3)/(m) x + (k)/(m)` |
`text(Infinite solutions) \ => \ text(gradients and y-intercepts equal)`
| `(m – 1)/(2)` | `= (3)/(m)` |
| `m^2 – m – 6` | `= 0` |
| `(m – 3)(m + 2)` | `= 0` |
`m = 3 \ \ text(or)\ \ –2`
`text(If) \ \ m = 3 ,`
`(k)/(3) = 1 \ => \ k = 3`
`text(If) \ \ m = –2,`
`(k)/(–2) = 1 \ => \ k = –2`
`=> \ D`
The simultaneous linear equations
`ax + 3y = 0`
`2x + (a + 1)y = 0`
where `a` is a real constant, have infinitely many solutions for
`B`
`text(Infinite solution:)`
| `m_1` | `= m_2` | `and` | `c_1` | `= c_2` |
| `a/2` | `= 3/(a + 1)` | `text(always true as both)` | ||
| `a` | `= – 3, 2` | `text(lines have)\ y text(-int) = 0` | ||
`=> B`
The simultaneous linear equations `(m - 1) x + 5y = 7 and 3x + (m - 3) y = 0.7 m` have infinitely many solutions for
`D`
`text(Infinite solutions occur when:)`
`->\ text(same gradient and same)\ \ y text(-intercept)`
| `(m_1 – 1) x + 5y_1` | `=7` |
| `y_1` | `=- (m_1 – 1)/5 x+ 7/5` |
| `3x + (m_2 – 3) y_2` | `= 0.7m` |
| `y_2` | `=- 3/(m_2 – 3)x +(0.7m)/(m_2 – 3)` |
`text(Gradients equal when,)`
| `- (m – 1)/5` | `= – 3/(m – 3)` |
| `m^2-4m+3` | `=15` |
| `(m-6)(m+2)` | `=0` |
`m = – 2 or m = 6`
`text(Coefficients equal when:)`
| `7/5` | `=(0.7m)/(m – 3)` |
| `7m – 21` | `= 3.5m` |
| `3.5m` | `=21` |
| `m` | `=6` |
`=> D`
Consider the simultaneous linear equations
| `kx - 3y` | `= k + 3` |
| `4x + (k + 7) y` | `= 1` |
where `k` is a real constant.
a. `text(Infinite solutions if gradients and)`
`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:)`
`m_1 != m_2`
`:. k in R\ text(\)\ {– 4, – 3}`