Problem Solving

In the xy coordinate system, if (a,b) and (a+3, b+k) are two points on the line defined by the equation x=3y-7, then k =

A. 9

B. 3

C. 7/3

D. 1

E. 1/3

Answer is D

How do I approach this problem? Also, could someone tell me how to tell if two lines are parallel?

Thanks so much!

gmatpup wrote:In the xy coordinate system, if (a,b) and (a+3, b+k) are two points on the line defined by the equation x=3y-7, then k =
A. 9
B. 3
C. 7/3
D. 1
E. 1/3
How do I approach this problem?

Method 1: Using the slope formula! The equation of any line can be represented in the form y = mx + c where m is the slope and c is the y intercept

x=3y-7 => y = (x+7)/3 = (1/3)*x + (7/3) = mx + c. So the slope of the line m = 1/3
To find the slope, you can also use the following formula:
Slope m = (y2-y1)/(x2-x1) where (x1,y1) and (x2,y2) are points that lie on the line y = mx + c
Slope m = (b+k-k)/(a+3-a) = k/3 = 1/3 => k = 1

Method 2: Substitution

We know x=3y-7 and points (a,b),(a+3,b+k) lie on the line.
So, substituting the values of (a,b)and (a+3,b+k) in the equation x=3y-7 we get
a = 3b - 7 => a -3b = -7 - (1)
a + 3 = 3b + 3k -7 => a = 3b -3 + 3k -7 => a -3b = -3 + 3k -7 - (2)
Equating the terms of right hand side of equations (1) and (2), you get
-7 = -3 + 3k -7 => 3k = 3 => k = 1

IMO D

Also, could someone tell me how to tell if two lines are parallel?

Since slope is a measure of the angle of a line from the horizontal, and since parallel lines must have the same angle, the parallel lines have the same slope - and lines with the same slope are parallel.

If lines y = mx + c and y = nx + d are parallel then the value of m and n should be equal

Doesn't this mean that the two lines are part of x=3y-7?

I just solved for y and then plugged in numbers.


Since we know y= x/3 + 7/3

I made x = 2 and then solved for the problem if x was +3, which resulted in y making a change from 3 to 4 equal to 1, the change of y.