Question

How many Solutions does this system have? (1 point)

2x+y=3
6x=9-3y

A)one
B)none
C)infinite
D)two

Answer 1

infinite

Explanation:

Answer 2

The given system of equation that is
`2x+y=3` and
`6x=9-3y` has infinite number of solutions.

Option -C.

Solution:

Need to determine number of solution given system of equation has.

`\begin{array}{l}{2 x+y=3} \\\\ {6 x=9-3 y}\end{array}`

Let us first bring the equation in standard form for comparison

`\begin{array}{l}{2 x+y-3=0} \\\\ {6 x+3 y-9=0}\end{array}`

`(a_(1))/(a_(2))=(b_(1))/(b_(2)) \\eq (c_(1))/(c_(2))`

To check how many solutions are there for system of equations
`a_(1) x+b_(1) y+c_(1)=0 \text{ and }a_(2) x+b_(2) y+c_(2)=0`, we need to compare ratios of
`(a_(1))/(a_(2)), (b_(1))/(b_(2)) \text { and } (c_(1))/(c_(2))`

In our case,

`a_(1) = 2, b_(1)= 1\text{ and }c_(1)= -3`

`a_(2)  = 6, b_(2) = 3,\text{ and }c_(2) = -9`

`\begin{array}{l}{\Rightarrow (a_(1))/(a_(2))=(2)/(6)=(1)/(3)} \\\\ {\Rightarrow (b_(1))/(b_(2))=(1)/(3)} \\\\ {\Rightarrow (c_(1))/(c_(2))=(-3)/(-9)=(1)/(3)} \\\\ {\Rightarrow (a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))=(1)/(3)}\end{array}`

As
`(a_(1))/(a_(2))=(b_(1))/(b_(2))=(c_(1))/(c_(2))`, so given system of equations have infinite number of solutions.

Hence, we can conclude that system has infinite number of solutions.