Question

The system of equations below is consistents with infinity many solutions where a and b are constants.Find the values of a and b. 2x+y=6 ax+by=9

Answer 1

Answer:

The values of $a$ and $b$ so that the two linear equations have infinite solutions are $3$ and $\frac{3}{2}$, respectively.

Step-by-step explanation:

Two linear equations with two variables have infinite solution if and only if they are linearly dependent. That is, one linear equation is a multiple of the other one. Let be the following system of linear equations:

$2\cdot x + y = 6$ (1)

$a\cdot x + b\cdot y = 9$ (2)

The following condition must be observed:

$r = \frac{a}{2} = \frac{b}{1} = \frac{9}{6}$ (3)

After some quick operations, we find the following information:

$r = \frac{3}{2}$, $a = 3$, $b = \frac{3}{2}$

The values of $a$ and $b$ so that the two linear equations have infinite solutions are $3$ and $\frac{3}{2}$, respectively.