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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.
Infinitely many solutions.
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 |
|---|---|---|---|
| 2y=16 x -2 | y= 8x+( - 1) | 8 | (0, - 1) |
| y=8 x -1 | y= 8x+( - 1) | 8 | (0, - 1) |
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 the same because the point at which each line crosses the y-axis is different. This means that both equations describe the same line and the system has infinitely many solutions.