In the xy-plane, line m has a slope of 2 and crosses the x-axis at the point (5/2, 0). Line n is perpendicular to line m and crosses the y-axis at the point (0, -5/2). At what point do lines m and n intersect?
A) (5/2, -5/2)
B) (4, -2)
C) (1, -3)
D) (2, -5/2)
To find the point at which lines m and n intersect, we need to determine the equations of the two lines.
Since line m has a slope of 2 and crosses the x-axis at (5/2, 0), we can find its equation using the point-slope form: y - y1 = m(x - x1), where (x1, y1) is the given point on the line.
Using the point (5/2, 0) and slope m = 2, the equation of line m is:
y - 0 = 2(x - 5/2)
y = 2x - 5
Line n is perpendicular to line m, so its slope will be the negative reciprocal of 2, which is -1/2. Since line n crosses the y-axis at (0, -5/2), we can find its equation using the same point-slope form.
Using the point (0, -5/2) and slope m = -1/2, the equation of line n is:
y - (-5/2) = -1/2(x - 0)
y + 5/2 = -1/2x
y = -1/2x - 5/2
Now we have the equations of lines m and n. To find the point of intersection, we need to solve these two equations simultaneously.
Setting the two equations equal to each other, we have:
2x - 5 = -1/2x - 5/2
Multiplying through by 2 to eliminate fractions:
4x - 10 = -x - 5
Combining like terms:
5x = 5
x = 1
Now substitute this value of x into either equation (let's use line m), to find the value of y:
y = 2(1) - 5
y = 2 - 5
y = -3
Therefore, the point at which lines m and n intersect is (1, -3).
The correct answer is C) (1, -3).
Since line m has a slope of 2 and crosses the x-axis at (5/2, 0), we can write the equation of line m in point-slope form as y - 0 = 2(x - 5/2).
Simplifying this equation, we get y = 2x - 5.
Since line n is perpendicular to line m, the slope of line n is the negative reciprocal of the slope of line m, which is -1/2.
Since line n crosses the y-axis at (0, -5/2), we can write the equation of line n in point-slope form as y - (-5/2) = -1/2(x - 0).
Simplifying this equation, we get y = -1/2x - 5/2.
To find the point of intersection of line m and line n, we can solve the system of equations:
y = 2x - 5 (equation of line m)
y = -1/2x - 5/2 (equation of line n)
Setting the two equations equal to each other, we get:
2x - 5 = -1/2x - 5/2.
Adding 1/2x to both sides and adding 5/2 to both sides, we get:
2x + 1/2x = -5/2 + 5/2.
Combining like terms, we get:
5/2x = 0.
Dividing both sides by 5/2, we get:
x = 0.
Plugging this value of x back into either equation, we can solve for y:
y = 2(0) - 5,
y = -5.
Therefore, the point of intersection of line m and line n is (0, -5).
Thus, the answer is not listed.