Question

which equation could be the second equation in the system ?

Answer 1

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:

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

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

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

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

$3x+4y=12$

are:

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

Since:

$(0,3)\ne(0,6),$

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:

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

Since:

$(0,3)\ne(0,6),$

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.