Question

which statement about the following system is correct?
$3y = -6x + 12$
$2x + y = 8$
\bigcirc the system is consistent because it has a solution.
\bigcirc the system is inconsistent because the two equations actually represent the same line.
\bigcirc the system is consistent because it has no solution.
\bigcirc the system is inconsistent because it has no solution.
the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.

Explanation:

Step1: Simplify the first equation

We have the first equation \(3y = -6x + 12\). Divide both sides by 3:
\(y=\frac{-6x + 12}{3}=-2x + 4\)

Step2: Simplify the second equation

The second equation is \(2x + y = 8\). Solve for \(y\):
\(y=-2x + 8\)

Step3: Analyze the slopes and intercepts

The first equation \(y = -2x+4\) has a slope of \(-2\) and a y - intercept of \(4\). The second equation \(y=-2x + 8\) has a slope of \(-2\) (same as the first equation) and a y - intercept of \(8\) (different from the first equation). Parallel lines (same slope, different y - intercepts) never intersect, so the system of equations has no solution. A system of equations with no solution is inconsistent. Wait, no, wait. Wait, let's re - check. Wait, maybe I made a mistake. Wait, let's rewrite the first equation correctly.

Wait, the first equation: \(3y=-6x + 12\), divide by 3: \(y=-2x + 4\). The second equation: \(2x + y=8\), so \(y=-2x + 8\). These are two parallel lines (same slope, different y - intercepts), so they never meet, so the system has no solution. But wait, the options: Wait, no, wait, maybe I misread the first equation. Wait, let's check again. Wait, the first equation is \(3y=-6x + 12\), let's rearrange it: \(6x+3y = 12\), divide by 3: \(2x + y=4\). The second equation is \(2x + y=8\). Ah! Now I see my mistake. So the first equation can be rewritten as \(2x + y=4\) (by adding \(6x\) to both sides of \(3y=-6x + 12\) to get \(6x + 3y=12\), then dividing by 3: \(2x + y = 4\)). The second equation is \(2x + y=8\). So we have the system:

\(\begin{cases}2x + y=4\\2x + y=8\end{cases}\)

Subtract the first equation from the second equation: \((2x + y)-(2x + y)=8 - 4\), which gives \(0 = 4\), which is a contradiction. So the system has no solution. A system of linear equations is consistent if it has at least one solution, and inconsistent if it has no solutions. So the system is inconsistent because it has no solution. But wait, the options: Let's check the options again.

Wait, the options are:

1. The system is consistent because it has a solution. (Wrong, since no solution)

2. The system is inconsistent because the two equations actually represent the same line. (Wrong, because we saw they are different lines)

3. The system is consistent because it has no solution. (Wrong, consistent means has solution)

4. The system is inconsistent because it has no solution. (This is correct)

Wait, but wait, earlier when I first simplified, I made a mistake in rearranging the first equation. Let's do it properly:

Starting with \(3y=-6x + 12\), add \(6x\) to both sides: \(6x+3y = 12\), divide both sides by 3: \(2x + y=4\). The second equation is \(2x + y=8\). So we have two equations \(2x + y = 4\) and \(2x + y=8\). If we subtract the first equation from the second: \((2x + y)-(2x + y)=8 - 4\Rightarrow0 = 4\), which is false. So the system has no solution, so it is inconsistent. But wait, the third option says "The system is inconsistent because the two equations actually represent the same line" which is wrong. The fourth option says "The system is inconsistent because it has no solution" which is correct. Wait, but let's check the original equations again. Wait, maybe I misread the first equation. Let's check the original problem again:

First equation: \(3y=-6x + 12\)

Second equation: \(2x + y=8\)

Let's solve the system using substitution. From the first equation, \(y=\frac{-6x + 12}{3}=-2x + 4\). Substitute into the second equation: \(2x+(-2x + 4)=8\Rightarrow2x-2x + 4=8\Rightarrow4 = 8\), which is a contradiction. So the system has no solution, so it is inconsistent. So the correct option is "The system is inconsistent because it has no solution."

Answer:

The system is inconsistent because it has no solution. (The option corresponding to this statement)