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: Analyze the first equation

Rewrite \(6x + y = -30\) in slope - intercept form (\(y=mx + b\), where \(m\) is the slope and \(b\) is the \(y\) - intercept).
Subtract \(6x\) from both sides: \(y=-6x - 30\). So the slope \(m_1=-6\) and the \(y\) - intercept \(b_1=-30\).

Step2: Analyze the second equation

Rewrite \(36x + 6y = 24\) in slope - intercept form.
First, subtract \(36x\) from both sides: \(6y=-36x + 24\).
Then divide each term by 6: \(y=-6x + 4\). So the slope \(m_2=-6\) and the \(y\) - intercept \(b_2 = 4\).

Step3: Compare slopes and \(y\) - intercepts

• For the slope: Since \(m_1=m_2=-6\) and \(b_1

eq b_2\) (\(-30
eq4\)), the two lines are parallel.

• For the number of solutions: Parallel lines (with different \(y\) - intercepts) have no solution. So the statement "This system has one solution" is false.
• For the true statement after solving: If we try to solve the system \(

$$\begin{cases}y=-6x - 30\\y=-6x + 4\end{cases}$$

\), substituting \(y\) gives \(-6x-30=-6x + 4\). Subtract \(-6x\) from both sides: \(-30 = 4\), which is a false statement. So the statement "After solving this system, you get a true statement" is false.

• For the same \(y\) - intercept: Since \(b_1=-30\) and \(b_2 = 4\), the \(y\) - intercepts are different. So the statement "The equations in this system have the same \(y\) - intercept" is false.

Answer:

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