Line A passes through the points (-1, 6) and (4, 14). Line B passes through the points (5, 17) and (3, 9). Which statement is true? Line A does not intersect line B, Line A interse... Line A passes through the points (-1, 6) and (4, 14). Line B passes through the points (5, 17) and (3, 9). Which statement is true? Line A does not intersect line B, Line A intersects line B at exactly one point, Line A overlaps line B.
Understand the Problem
The question is asking which statement is true about two lines, A and B, based on the points they pass through. We need to determine their intersection or overlap.
Answer
Line A intersects line B at exactly one point.
Answer for screen readers
Line A intersects line B at exactly one point.
Steps to Solve
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Determine the slope of Line A
Line A passes through the points ((-1, 6)) and ((4, 14)). The formula for the slope (m) between two points ((x_1, y_1)) and ((x_2, y_2)) is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Plugging in our points:
$$ m_A = \frac{14 - 6}{4 - (-1)} = \frac{8}{5} $$
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Write the equation of Line A
Using the slope-intercept form (y = mx + b), where (m) is the slope and (b) is the y-intercept. We can use one of the points, say ((-1, 6)), to find (b):
$$ 6 = \frac{8}{5}(-1) + b $$
Solving for (b):
$$ 6 = -\frac{8}{5} + b \ b = 6 + \frac{8}{5} = \frac{30}{5} + \frac{8}{5} = \frac{38}{5} $$
Therefore, the equation for Line A is:
$$ y = \frac{8}{5}x + \frac{38}{5} $$
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Determine the slope of Line B
Line B passes through the points ((5, 17)) and ((3, 9)). We can calculate the slope similarly:
$$ m_B = \frac{9 - 17}{3 - 5} = \frac{-8}{-2} = 4 $$
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Write the equation of Line B
Using the slope-intercept form for Line B and the point ((5, 17)):
$$ 17 = 4(5) + b $$
Solving for (b):
$$ 17 = 20 + b \ b = 17 - 20 = -3 $$
Therefore, the equation for Line B is:
$$ y = 4x - 3 $$
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Find the intersection of Line A and Line B
Set the equations equal to each other to find the point of intersection:
$$ \frac{8}{5}x + \frac{38}{5} = 4x - 3 $$
Multiply through by 5 to eliminate the fraction:
$$ 8x + 38 = 20x - 15 $$
Rearranging gives:
$$ 12x = 53 \ x = \frac{53}{12} $$
Substitute (x) back into one of the equations to find (y) (using Line A):
$$ y = \frac{8}{5}\left(\frac{53}{12}\right) + \frac{38}{5} $$
Simplifying yields:
$$ y = \frac{424}{60} + \frac{456}{60} = \frac{880}{60} \approx 14.67 $$
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Identify relationship between the lines
Since there is an intersection point, Line A intersects Line B at exactly one point.
Line A intersects line B at exactly one point.
More Information
This implies that the two lines are not parallel and will cross each other, confirming the uniqueness of the intersection.
Tips
- Failing to properly calculate the slope can lead to incorrect conclusions about the relationship between the lines.
- Not substituting back correctly into the equation to find the coordinates of the intersection point.
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