Question

How do you tell whether the system $ x + 2y = 10 $ and $ 2x + 4y = 10 $ has no solution or infinitely many solutions?

Answer 1

Hint: As we know that the above given equations are examples of linear equations. An equation for a straight line is called a linear equation. The standard form of linear equations in two variables is $ Ax + By = C $ . Here we will use the substitution method to solve this equation or we can compare the coefficients of the two equations and then solve it.

Complete step-by-step answer:
As we know that the above given equation is a linear equation. Two linear equations are written as
 $ {a_1}x + {b_1}y + {c_1} = 0 $ and $ {a_2}x + {b_2}y + {c_2} = 0 $ .
Now both the equations can be written as
 $ x + 2y - 10 = 0 $ and $ 2x + 4y - 10 = 0 $ .
By comparing both the equations we have
 $ {a_1} = 1,{b_1} = 2,{c_1} = - 10 $ and $ {a_2} = 1,{b_2} = 4,{c_2} = - 10 $ .
We know that if
 $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ then the lines are parallel and they have no solutions at all.
By comparing the equations we have
 $ \dfrac{1}{2} = \dfrac{2}{4} \ne \dfrac{{ - 10}}{{ - 10}} \Rightarrow \dfrac{1}{2} = \dfrac{1}{2} \ne \dfrac{1}{1} $ .
Hence the given set of equations have no solutions at all.

Note: We should keep in mind the positive and negative signs while calculating the value of any variable. Since the lines are parallel with no point of intersection hence there is no solution. We should also know that if there is $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $ , then the equations have infinitely many solution and if there is $ \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} $ , then the equations have unique solution.