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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.
One solution.
To do this, use the slope-intercept form of each equation, where m is the slope and the point (0, b) is the y-intercept. There are three possibilities when comparing two linear equations in a system.
| 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 = 4x+8 | y= 4x+ 8 | 4 | (0, 8) |
| y= 5x+1 | y= 5x+ 1 | 5 | (0, 1) |
By comparing the slopes we see that they are different, so the lines are not parallel. This means that the lines intercept at some point and the system has exactly one solution.