Question

What value of b will cause the system to have an infinite

number of solutions?
y = 6xb
-3x +
y = -3

Answer 1

For infinite many solution , b =
`(1)/(2)`

Explanation:

Given two linear equation as,

- 6bX + Y +0 =0 and

-3X + Y + 3 =0

Here given that system have infinite number of solutions

Now for infinite number of solutions ,

`(a1)/(b1)` =
`(b1)/(b2)` =
`(c1 )/(c2)`

Thus from given linear equation ,the coefficient of both equation be ,

a1 = -6b b1 = 1 c1 = 0 for -6bX + Y + 0 = 0

a2 = -3 b2 = 1 c2 =3 for -3X + Y + 3 = 0

SO, from the condition of infinite solutions

`(-6b)/(-3)` =
`(1)/(1)` =
`(0)/(3)`

∴ 6b = 3

So, b =
`(3)/(6)`

Hence b=
`(1)/(2)` Answer

Answer 2

A system of equations has infinite solutions when we have more variables than linear independent equations.

This means that if we want to have infinite solutions, we need to find a value of b such the two equations are linear dependent, his means that the equations are the same, or that one equation is a scalar times the other, like x + y = 2. and 2x + 2y = 4 (the second is 2 times the first one)

Here we have the equations:

y = 6xb - 3x

y = -3x

(the second equation actually says y = -3, but this may be written wrong)

So we want that both equations are the same (because in both cases we have y equals something), for this, we need to find the value of b:

6xb - 3x = -3x

6xb = - 3x + 3x = 0

and this must work for every value of x, so we must have that b = 0.