Question

A system of equations has infinitely many solutions. If 2y-4x=6 is one of the equations, which could be the other equation?

Answer 1

"infinity many solutions" implies that the two lines coincide.

Example: starting with 2y-4x=6, I multiply every term by 3: 6y-12x=18

These two different-appearing equations are mathematically identical, so graphing both on the same set of axes results in two lines that coincide.

Answer 2

Equation is
`-8x+4y-12=0`

Explanation:

An equation is a mathematical statement that two things are equal .

A system of equations is a set of two or more equations consisting of same unknowns.

Two equations
`a_1x+b_1y+c_1=0\,,\,a_2x+b_2y+c_2=0` have unique solution if
`(a_1)/(a_2)\\eq (b_1)/(b_2)`

infinite solution if
`(a_1)/(a_2)= (b_1)/(b_2)=(c_1)/(c_2)`

no solution if
`(a_1)/(a_2)= (b_1)/(b_2)\\eq (c_1)/(c_2)`

Here, given:
`-4x+2y=6`

We can write this equation as
`-4x+2y-6=0`

Take another equation as
`-8x+4y-12=0`

Here,
`a_1=-4\,,\,a_2=-8\,,\,b_1=2\,,\,b_2=4\,,\,c_1=-6\,,\,c_2=-12`

`(a_1)/(a_2)=(-4)/(-8)=(1)/(2)\\(b_1)/(b_2)=(2)/(4)=(1)/(2)\\(c_1)/(c_2)=(-6)/(-12)=(1)/(2)`

such that
`(a_1)/(a_2)=(b_1)/(b_2)=(c_1)/(c_2)`