Question

I'm studying for my algebra exam and I need help with a specific problem: finding the graph that represents the solution to the system of equations −3x−y=3 and y=−3x+4. Can you find the correct graph for me?

Answer 1

To find the graph that represents the solution to the system of equations given by \(-3x - y = 3\) and \(y = -3x + 4\), let's start by rewriting the first equation in slope-intercept form (\(y = mx + b\)).

1. Starting with the first equation:
\[
-3x - y = 3
\]
Add \(3x\) to both sides:
\[
-y = 3 + 3x
\]
Multiply by -1 to solve for \(y\):
\[
y = -3x - 3
\]

Now we have two equations:
1. \(y = -3x - 3\)
2. \(y = -3x + 4\)

Next, we can plot both equations on a graph:

### Step 1: Plotting the first equation \(y = -3x - 3\)
- The y-intercept is \(-3\) (when \(x = 0\), \(y = -3\)).
- The slope is \(-3\), which means for every 1 unit increase in \(x\), \(y\) decreases by 3 units.

### Step 2: Plotting the second equation \(y = -3x + 4\)
- The y-intercept is \(4\) (when \(x = 0\), \(y = 4\)).
- This line also has a slope of \(-3\), and it will decrease similarly as the first line.

### Step 3: Finding the Intersection
Since both lines have the same slope (\(-3\)), they are parallel lines. Parallel lines do not intersect, which means there is no solution to this system of equations.

### Final Answer:
- The correct graph for this system will show two parallel lines:
- One line represented by \(y = -3x - 3\)
- One line represented by \(y = -3x + 4\)

Since these lines are parallel, there will be no point of intersection, indicating that there is no solution to this system. Ensure you select or draw a graph that visually represents this situation correctly.