To solve the system of equations by graphing, we need to first express both equations in slope-intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
The first equation is already in slope-intercept form:
- \(y = -3x + 4\)
For the second equation \(y + 3x = -4\), we will rearrange it to isolate \(y\):
\[ y = -3x - 4 \]
Now we have the two equations ready to graph:
- \(y = -3x + 4\)
- \(y = -3x - 4\)
Graphing the Equations
For the first equation \(y = -3x + 4\):
- The y-intercept (where \(x = 0\)) is \(4\). This gives the point \((0, 4)\).
- The slope is \(-3\). This means that for each increase of \(1\) in \(x\), \(y\) decreases by \(3\). Using the y-intercept point:
- When \(x = 1\), \(y = -3(1) + 4 = 1\) → Point \((1, 1)\).
- When \(x = -1\), \(y = -3(-1) + 4 = 7\) → Point \((-1, 7)\).
For the second equation \(y = -3x - 4\):
- The y-intercept is \(-4\). This gives the point \((0, -4)\).
- The slope is also \(-3\). So we can find additional points:
- When \(x = 1\), \(y = -3(1) - 4 = -7\) → Point \((1, -7)\).
- When \(x = -1\), \(y = -3(-1) - 4 = -1\) → Point \((-1, -1)\).
Plotting
Now, we can plot the points calculated for each equation:
-
For \(y = -3x + 4\):
- Points: \((0, 4)\), \((1, 1)\), \((-1, 7)\)
-
For \(y = -3x - 4\):
- Points: \((0, -4)\), \((1, -7)\), \((-1, -1)\)
Intersection Point
Since both lines are parallel (same slope of \(-3\)), they will never intersect. This means there is no solution to this system of equations. The lines are parallel, and therefore do not share any points in common.
Thus, the final conclusion is that the system of equations has no solution.