To determine whether the system of equations \( y = 3x + 4 \) and \( -3x + y = 8 \) has a solution, we can manipulate the second equation into a form that allows for easier comparison.
The first equation is already in slope-intercept form:
\[ y = 3x + 4 \]
For the second equation \(-3x + y = 8\), we can rearrange it into slope-intercept form by solving for \(y\):
\[ y = 3x + 8 \]
Now we have both equations:
- \( y = 3x + 4 \)
- \( y = 3x + 8 \)
Next, we analyze these two equations. Both lines have the same slope (3), but different y-intercepts (4 and 8). This means that the lines are parallel and will never intersect. Since parallel lines do not meet, there is no solution to the system of equations.
Thus, the graph that shows these two lines, which are parallel and do not intersect, correctly represents the system of equations with no solution. The correct graph would display two parallel lines, one for each equation, indicating that the system does not have any points of intersection.
1 answer