In a rectangular coordinate system, straight-line \(k\) passes through points \((0, 0)\) and \((3, 2).\) Which of the

Vincen wrote: ↑

Mon May 18, 2020 6:33 am

In a rectangular coordinate system, straight-line \(k\) 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. \((–6, –4)\)

[spoiler]OA=E[/spoiler]

Source: Official Guide

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

Vincen wrote: ↑

Mon May 18, 2020 6:33 am

In a rectangular coordinate system, straight-line \(k\) 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. \((–6, –4)\)

[spoiler]OA=E[/spoiler]

Source: Official Guide

Since line k passes through points (0, 0) and (3, 2), its slope is (2 - 0) / (3 - 0) = 2/3. If a point is on line k, then the slope calculated using its coordinates and one of the given points should be 2/3, the slope of the line. Since (0, 0) is an “easier” point for calculations than (3, 2), we will use (0, 0). For choice A, the slope between (0, 0) and (9, 4) is (4 - 0) / (9 - 0) = 4/9. We see that this is not 2/3, so choice A is not the correct answer.

Notice that the slope between (0, 0) and (a, b) is b/a. Therefore, we can determine the slope of the line easily for the other four choices and determine which is on line k.

B. (4, 9)

slope = 9/4 ≠ 2/3

C. (-4, 6)

slope = 6/(-4) = -3/2 ≠ 2/3

D. (-6, -9)

slope = (-9)/(-6) = 3/2 ≠ 2/3

E. (-6, -4)

slope = (-4)/(-6) = 2/3

Answer: E