Respuesta :
Given:
The system of equations is
[tex]y=2x-5[/tex]
[tex]2y-4x=?[/tex]
To find:
The missing value for which the given system of equations have infinitely many solutions.
Solution:
Let the missing value be k.
We have,
[tex]y=2x-5[/tex]
[tex]2y-4x=k[/tex]
Taking all the terms on the left side, the given equations can be rewritten as
[tex]-2x+y+5=0[/tex]
[tex]-4x+2y-k=0[/tex]
The system of equations [tex]a_1x+b_1y+c_1=0[/tex] and [tex]a_2x+b_2y+c_2=0[/tex] have infinitely many solutions if
[tex]\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}[/tex]
We have,
[tex]a_1=-2,b_1=1,c_1=5[/tex]
[tex]a_2=-4,b_2=2,c_2=-k[/tex]
Now,
[tex]\dfrac{-2}{-4}=\dfrac{1}{2}=\dfrac{5}{-k}[/tex]
[tex]\dfrac{1}{2}=\dfrac{1}{2}=\dfrac{5}{-k}[/tex]
[tex]\dfrac{1}{2}=\dfrac{5}{-k}[/tex]
On cross multiplication, we get
[tex]-k=10[/tex]
[tex]k=-10[/tex]
Therefore, the missing value is -10.
