In a rectangular coordinate system, straight line \(k\)

AAPL wrote:Official Guide

In a rectangular coordinate system, straight like \(k\) passes through points \((0, 0)\) and \((3, 2)\). Which of the following are coordinates of a point on \(k\)?

A. \((9, 4)\)
B. \((4, 9)\)
C. \((-4, 6)\)
D. \((-6, -9)\)
E. \((-6, -4)\)

OA E

Key concept: If a point lies ON a line, then the coordinates of that point must SATISFY the equation of that line.

So, let's first find the equation of the line.
Slope = (y2 - y1)/(x2 - x1) = (2 - 0)/(3 - 0) = 2/3
The y-intercept = 0

So, the equation of the line, in slope y-intercept form, is: y = (2/3)x + 0 or just y = 2x/3

Now check each answer choice to see which coordinates satisfy the equation y = 2x/3

A. (9, 4)
Plug values into y = 2x/3 to get: 4 = (2)(9)/3
Simplify: 9 = 6
Doesn't work.
ELIMINATE A

B. (4, 9)
Plug values into y = 2x/3 to get: 9 = (2)(4)/3
Simplify: 9 = 8/3
Doesn't work.
ELIMINATE B

C. (–4, 6)
Plug values into y = 2x/3 to get: 6 = (2)(-4)/3
Simplify: 6 = -8/3
Doesn't work.
ELIMINATE C

D. (–6, –9)
Plug values into y = 2x/3 to get: -9 = (2)(-6)/3
Simplify: -9 = -4
Doesn't work.
ELIMINATE D

E. (–6, –4)
Plug values into y = 2x/3 to get: -6 = (2)(-4)/3
Simplify: -6 = -6
WORKS!!


Answer: E

Cheers,
Brent

AAPL wrote:Official Guide

In a rectangular coordinate system, straight like \(k\) passes through points \((0, 0)\) and \((3, 2)\). Which of the following are coordinates of a point on \(k\)?

A. \((9, 4)\)
B. \((4, 9)\)
C. \((-4, 6)\)
D. \((-6, -9)\)
E. \((-6, -4)\)

OA E

We see that the slope of line k is (2 - 0)/(3 - 0) = 2/3. If a point is on line k, the slope between the point and either of the given points should be 2/3 also. We see that (-6, -4) is on line k since (-4 - 0)/(-6 - 0) = (-4)/(-6) = 2/3.

Alternate Solution:

We see that the slope of line k is (2 - 0)/(3 - 0) = 2/3. Thus, the equation of line k is y = (2/3)x. Of the given points, (-6, -4) satisfies this equation; therefore this point is on line k.

Answer: E