smc-721-30-No solutions

Functions, MET1 2023 VCE SM-Bank 4

Consider the simultaneous equations below, where \(a\) and \(b\) are real constants.

\begin{aligned}
& (a+3) x+9 y=3 b\\
& 2 x+a y=5
\end{aligned}

Find the values of \(a\) and \(b\) for which the simultaneous equations have no solutions. (4 marks)

--- 8 WORK AREA LINES (style=lined) ---

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\(a=-6\ \text{and}\ b\neq -\dfrac{5}{2}\ \ \text{OR}\ \ a=3\ \text{and}\ b\neq 5\)

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(\(a+3)x+9y\)	\(=3b\)
\(y\)	\(=-\dfrac{a+3}{9}x+\dfrac{b}{3}\)	\(\dots (1)\)
\(\rightarrow \ m_1\)	\(=-\dfrac{a+3}{9}, \ c_1=\dfrac{b}{3}\)

\(2x+ay\)	\(=5\)
\(y\)	\(=-\dfrac{2}{a}x+\dfrac{5}{a}\)	\(\dots (2)\)
\(\rightarrow \ m_2\)	\(=-\dfrac{2}{a}, \ c_2=\dfrac{5}{a}\)

\(\text{No solution if }m_1=m_2,\ \text{and}\ c_1\neq c_2.\)

\(\rightarrow\ -\dfrac{a+3}{9}=-\dfrac{2}{a}\ \text{and }\dfrac{b}{3}\neq \dfrac{5}{a}\rightarrow\ b\neq \dfrac{15}{a}\)

\(-\dfrac{a+3}{9}=-\dfrac{2}{a}\)

\(a^2+3a-18\)	\(=0\)
\((a+6)(a-3)\)	\(=0\)
\(\therefore a=-6\ \)	\(\ \text{or}\ \)	\(\ a=3\)

\(\text{When }\ a=-6,\ b\neq \dfrac{15}{-6}\rightarrow\ b\neq -\dfrac{5}{2}\)

\(\text{OR}\)

\(\text{When }\ a=3,\ b\neq \dfrac{15}{3}\rightarrow\ b\neq 5\)

Algebra, MET2 2015 VCAA 21 MC

The graphs of `y = mx + c` and `y = ax^2` will have no points of intersection for all values of `m, c` and `a` such that

`a > 0 and c > 0`
`a > 0 and c < 0`
`a > 0 and c > -m^2/(4a)`
`a < 0 and c > -m^2/(4a)`
`m > 0 and c > 0`

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`D`

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`text(Intersect when:)`

`mx + c`	`= ax^2`
`ax^2 – mx – c`	`= 0`

`text(S)text(ince no points of intersection:)`

♦ Mean mark 37%.

`Delta`	`< 0`
`m^2 – 4a(−c)`	`< 0`
`m^2 + 4ac`	`< 0`

`text(Solve for)\ c:`

`:.\ c > (−m^2)/(4a),quada < 0`

`text(or)`

`c < (−m^2)/(4a),quada > 0`

`=> D`

Algebra, MET2 2014 VCAA 17 MC

The simultaneous linear equations `ax - 3y = 5` and `3x - ay = 8 - a` have no solution for

`a = 3`
`a = -3`
both `a = 3` and ` a = -3`
`a in R text(\{3})`
`a in R text(\[−3, 3])`

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`B`

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`text(No solution occurs when:)`

♦ Mean mark 50%.

`=> m_1 = m_2quadtext(and)quadc_1 != c_2`

`a/3`	`= 3/a`	`text(and)`	`-5/3`	`!= -((8-a)/a)`
`a`	`= ±3`		`5a`	`!= 24-3a`
			`a`	`!=3`

`:. a = −3`

`=> B`

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.

What are the values of `k` where no solutions exist? (3 marks)
What values of `k` do the simultaneous equations have a unique solution? (1 mark)

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

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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}`