Question
Show that the system of equations:
$\text{6x}+\text{5y}=11,\ \text{9x}+\frac{15}{2}\text{y}=21$
has no solutions.
$\text{6x}+\text{5y}=11,\ \text{9x}+\frac{15}{2}\text{y}=21$
has no solutions.
✓
Answer
$\text{6x}+\text{5y}=11,\ \text{9x}+\frac{15}{2}\text{y}=21$
$\text{6x}+\text{5y}-11=0,$
$\text{9x}+\frac{15}{2}\text{y}-21=0$
We know that,
The system of linear equations
$a_1x + b_1y + c_1 = 0, a_2x + b_2y + c_2 = 0.$
has a no solution if $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
$\frac{\text{a}_1}{\text{a}_2}=\frac{6}{9}=\frac{2}{3}$
$\frac{\text{b}_1}{\text{b}_2}=\frac{5}{\frac{15}{2}}=\frac{10}{15}=\frac{2}{3}$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-11}{-21}=\frac{11}{21}$
Clearly, $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
Hence the given system of equations has no solutions.
$\text{6x}+\text{5y}-11=0,$
$\text{9x}+\frac{15}{2}\text{y}-21=0$
We know that,
The system of linear equations
$a_1x + b_1y + c_1 = 0, a_2x + b_2y + c_2 = 0.$
has a no solution if $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
$\frac{\text{a}_1}{\text{a}_2}=\frac{6}{9}=\frac{2}{3}$
$\frac{\text{b}_1}{\text{b}_2}=\frac{5}{\frac{15}{2}}=\frac{10}{15}=\frac{2}{3}$
$\frac{\text{c}_1}{\text{c}_2}=\frac{-11}{-21}=\frac{11}{21}$
Clearly, $\frac{\text{a}_1}{\text{a}_2}=\frac{\text{b}_1}{\text{b}_2}\neq\frac{\text{c}_1}{\text{c}_2}$
Hence the given system of equations has no solutions.
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