Question

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution find it,

x − 2y = 8
5x − 10y = 10

Answer 1

In order to solve this problem we will compare the coefficient of the two equations such that if the two equations are ${a_1}x + {b_1}y + {c_1} = 0$, ${a_2}x + {b_2}y + {c_2} = 0$. Then we will compare the coefficients such that $\dfrac{{{a_1}}}{{{a_2}}}$, $\dfrac{{{b_1}}}{{{b_2}}}$ and $\dfrac{{{c_1}}}{{{c_2}}}$. If $\dfrac{{{a_1}}}{{{a_2}}}$ $\ne$ $\dfrac{{{b_1}}}{{{b_2}}}$ then the equations have unique solution, if $\dfrac{{{a_1}}}{{{a_2}}}$ = $\dfrac{{{b_1}}}{{{b_2}}}$ = $\dfrac{{{c_1}}}{{{c_2}}}$ then the equations have infinitely many solutions and if $\dfrac{{{a_1}}}{{{a_2}}}$ = $\dfrac{{{b_1}}}{{{b_2}}}$ $\ne$ $\dfrac{{{c_1}}}{{{c_2}}}$ then the equations have no solutions. Doing this will solve your problem and will give you the right answer.

Complete step-by-step answer :
The given equations are,
x − 2y = 8
5x − 10y = 10
The above equations can be written as,
x − 2y – 8 = 0
5x − 10y – 10 = 0
We need to find the conditions of number of solutions of the equations,
So, we know that if the two equations are ${a_1}x + {b_1}y + {c_1} = 0$, ${a_2}x + {b_2}y + {c_2} = 0$. Then we will compare the coefficients such that $\dfrac{{{a_1}}}{{{a_2}}}$, $\dfrac{{{b_1}}}{{{b_2}}}$ and $\dfrac{{{c_1}}}{{{c_2}}}$. If $\dfrac{{{a_1}}}{{{a_2}}}$ $\ne$ $\dfrac{{{b_1}}}{{{b_2}}}$ then the equations have unique solution, if $\dfrac{{{a_1}}}{{{a_2}}}$ = $\dfrac{{{b_1}}}{{{b_2}}}$ = $\dfrac{{{c_1}}}{{{c_2}}}$ then the equations have infinitely many solutions and if $\dfrac{{{a_1}}}{{{a_2}}}$ = $\dfrac{{{b_1}}}{{{b_2}}}$ $\ne$ $\dfrac{{{c_1}}}{{{c_2}}}$ then the equations have no solutions.
Here we can clearly see that,
${a_1} = 1,\,{b_1} = - 2,\,{c_1} = - 8 \\\ {a_2} = 5,\,{b_2} = - 10,\,{c_2} = - 10 \\\$
So, we do, $\dfrac{{{a_1}}}{{{a_2}}},\,\dfrac{{{b_1}}}{{{b_2}}},\,\dfrac{{{c_1}}}{{{c_2}}}$.
We see that, $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{1}{5},\,\dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{5},\,\dfrac{{{c_1}}}{{{c_2}}} = \dfrac{4}{5}$.
We can clearly see that $\dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} \Rightarrow \dfrac{1}{5} = \dfrac{1}{5} \ne \dfrac{4}{5}$.
So, the system of equations has no solution according to the information above.

Note : When you get to solve such problems you always need to consider the coefficients of linear equations and perform the operations as done above to get the correct result. Above equations are the equations of two lines and if the lines are parallel then it has no solution because they are not intersecting, if the lines are crossing each other then they have a unique solution and if the lines are coinciding then they have many solutions. Knowing this will solve your problem and will give you the right answers.