Mid-Chapter Quiz
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What do you notice about the equations of lines that intersect at one point? How about ones that never intersect? Ones that overlap entirely?
The system is consistent and independent.
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 | Classification |
|---|---|---|---|
| m_1 ≠ m_2 | irrelevant | intersecting lines | consistent, independent |
| m_1=m_2 | b_1≠ b_2 | parallel lines | inconsistent |
| m_1=m_2 | b_1=b_2 | same line | consistent, dependent |
Let's write 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=2x-1 | y=2x+( -1) | 2 | (0, -1) |
| y=-2x+3 | y=-2x+ 3 | -2 | (0, 3) |
Comparing the slopes, we see that they are not equal, so the lines are intersecting. This means that the system is consistent and independent.