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If the lines have different slopes, then there is only one solution. If they have the same slope but different y-intercepts, then there is no solution. Finally, if the lines have the same slope and the same y-intercept, then there are infinitely many solutions.
No solution.
| Slope | y-intercept | Graph Description | Number of Solutions |
|---|---|---|---|
| m_1≠ m_2 | Irrelevant | Intersecting lines | One solution |
| m_1=m_2 | b_1≠ b_2 | Parallel lines | No solution |
| m_1=m_2 | b_1=b_2 | Same line | Infinitely many |
Let's rewrite the equations in the given system in slope-intercept form, highlighting the m and b values.
| Given Equation | Slope-Intercept Form | Slope m | y-intercept b |
|---|---|---|---|
| y=- 3x+2 | y= - 3x+ 2 | - 3 | (0, 2) |
| y=- 3x+5 | y= - 3x+ 5 | - 3 | (0, 5) |
By comparing the slopes we see that they are equal, so the lines are either parallel or the same. Looking at the y-intercepts, we can tell the lines are different because the point at which each line crosses the y-axis is different. This means that the lines are parallel and the system has no solution.