In the xy-coordinate system, if (m, n) and (m 1 2, n 1 k) are two points on the line
with the equation x 5 2y 1 5, then k 5
Choose the correct answer:
A) 1/2
B) 1
C) 2
D) 5/2
E) 4
Step 1: Analyze the Question   For any question involving the equation of a line, a good   place to start is the slope-intercept form of the line,   y = mx 1 b. Remember that if you have two points on a   line, you can derive the entire equation, and if you have an   equation of the line, you can calculate any points on that   line.   Step 2: State the Task   We are solving for k, which is the amount by which the   y-coordinate increases when the x-coordinate increases   by 2.   Step 3: Approach Strategically   The slope of a line is the ratio between the change in y and   the change in x. In other words, every time the x-coordinate   increases by 1, the y-coordinate increases by the amount   of the slope.   The equation of the line in the question stem is defined as   x = 2y + 5. We must isolate y to have slope-intercept form:   So the slope of this line is 1/2 . This means that for every   change of +1 in the x direction, there is a change of + 1/2   in the y direction. Then we know that, because there is an   increase in 2 units in the x direction when moving from   m to m + 2, there must be a change of 1 unit in the y   direction when moving from n to n + k. So k = 1.   Since there are variables that eventually cancel (m and n   are not part of the answers), we can Pick Numbers. Let’s   say that you choose the y-coordinate of the point (m, n) to   be 0 to allow for easier calculations. Using the equation   we’re given to relate x- and y-coordinates, we can calculate   the x-coordinate:   So (m, n) is the point (5, 0).   Now we’ll plug our values of m and n into the next point:   (m + 2, n + k). That yields (7, k). All we have to do is plug   an x-coordinate of 7 into the equation to solve for k, the   y-coordinate:   Answer is 1
<result>B</result>
In the xy-coordinate system, if (m, n) and (m 1 2, n 1 k) are two points on the line
with the equation x 5 2y 1 5, then k 5