Question

Which equation could be the second equation in the system ?

Answer 1

3x + 4y= 12

Explanation:

if I’m not mistaken this equation will be under the other one cause no solutions but I will check desmos and comment is correct

Answer 2

Recall that a system of two equations with two variables has no solutions if the equations have the same slope but a different y-intercept.

Also, recall that the slope and the y-intercept of the graph of a linear equation in general form:

`Ax+By=C`

are:

`\begin{gathered} \text{slope}=-(A)/(B), \\ y-\text{intercept}=(0,(C)/(B)_{})\text{.} \end{gathered}`

Therefore, the slope and the y-intercept of the given equation are:

`\begin{gathered} \text{slope}=-(3)/(4), \\ y-\text{intecept}=(0,(24)/(4))=(0,6)\text{.} \end{gathered}`

Now, notice that the slope and the y-intercept of the equation:

`3x+4y=12`

are:

`\begin{gathered} \text{slope}=-(3)/(4), \\ y-\text{intercept}=(0,(12)/(4))=(0,3)\text{.} \end{gathered}`

Since:

`(0,3)\\e(0,6),`

we get that the system of equations:

`\begin{cases}3x+4y=24 \\ 3x+4y=12\end{cases}\text{.}`

has no solutions.

Also, notice that the slope and the y-intercept of the equation:

`-3x-4y=-12`

are:

`\begin{gathered} \text{slope}=-(-3)/(-4)=-(3)/(4), \\ y-\text{intercept}=(0,(-12)/(-4))=(0,3)\text{.} \end{gathered}`

Since:

`(0,3)\\e(0,6),`

we get that the system of equations:

`\begin{cases}3x+4y=24 \\ -3x-4y=-12\end{cases}\text{.}`

has no solutions.

Answer: First and third options are both correct.