Question

examine the system of equations.
6x + y = -30,
36x + 6y = 24
which statement is true about the system of equations?
○ after solving this system, you get a true statement.
○ this system has one solution.
○ if this system were graphed, the lines would be parallel.
○ the equations in this system have the same y-intercept.

Explanation:

Step1: Rewrite equations in slope - intercept form

For the first equation \(6x + y=-30\), we can rewrite it as \(y=-6x - 30\). The slope \(m_1=-6\) and the \(y\) - intercept \(b_1=-30\).

For the second equation \(36x + 6y = 24\), we first solve for \(y\):

$$
LATEXBLOCK0
$$

The slope \(m_2=-6\) and the \(y\) - intercept \(b_2 = 4\).

Step2: Analyze the slopes and intercepts

• Slope analysis: Since \(m_1=m_2=-6\) and \(b_1

eq b_2\) (\(-30
eq4\)), the two lines are parallel. Parallel lines do not intersect, so the system of equations has no solution.

• Checking other options:
• Since the system has no solution, when we try to solve it (for example, by substitution or elimination), we will get a false statement, so the option "After solving this system, you get a true statement" is wrong.
• Since the lines are parallel and non - coincident, the system has no solution, so the option "This system has one solution" is wrong.
• The \(y\) - intercepts are \(-30\) and \(4\) which are not the same, so the option "The equations in this system have the same \(y\) - intercept" is wrong.

Answer:

If this system were graphed, the lines would be parallel.