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Question

$table attributes columnspacing 1em end attributes row cell 3 over 4 x plus 1 half y equals 12 end cell row cell a x minus b y equals 9 end cell end table$
The system of equations above has no solutions. If a and b are constants, what is the value of $a over b$ ?

The correct answer is: 3/2

○        The concept used in this question is the no of solutions of pair of equations.
If a₁x + b₁y + c₁= 0 and a₂x + b₂y + c₂ = 0 are two pair of equation then equations has
Unique solution if   $a subscript 1 over a subscript 2 not equal to b subscript 1 over b subscript 2$
No solution if $a subscript 1 over a subscript 2 equals b subscript 1 over b subscript 2 not equal to c subscript 1 over c subscript 2$
infine solution if   $a subscript 1 over a subscript 2 equals b subscript 1 over b subscript 2 equals c subscript 1 over c subscript 2$
●        Step by step explanation:
Given:
$3 over 4 x plus 1 half y equals 12$
ax - by = 9
has no solutions.
○        Step 1:
○        Compare equation with general equation ax + by + c = 0.

We get,
$straight a subscript 1 equals 3 over 4 comma straight b subscript 1 equals 1 half comma straight c subscript 1 equals negative 12$
a₂= a, b₂ = -b, c₂ = -9
○        Step 2:
Condition for no solution.

$a subscript 1 over a subscript 2 equals b subscript 1 over b subscript 2 not equal to c subscript 1 over c subscript 2$

Now, put values in it.
$not stretchy rightwards double arrow fraction numerator 3 over 4 over denominator a end fraction equals fraction numerator 1 half over denominator b end fraction not equal to fraction numerator negative 12 over denominator negative 9 end fraction$
Now cross multiply,
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow fraction numerator 3 over 4 over denominator 1 half end fraction equals a over bold italic b end cell row cell not stretchy rightwards double arrow 3 over 4 cross times 2 over bold 1 equals a over bold italic b end cell end table$
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell not stretchy rightwards double arrow 6 over bold 4 to the power of b equals a over bold italic b end cell row cell not stretchy rightwards double arrow 3 over 2 equals a over bold italic b end cell end table$
$therefore a over bold italic b equals 3 over 2.$
●        Final Answer:
Hence, value of
$a over bold italic b text is end text 3 over 2$

infine solution if

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