Intermath | Workshop Support


Comparing Lines


Problem Statement
 Let f(x) = ax + b, and g(x) = cx + d, where a, b, c, and d, are any real numbers.
If f(x) and g(x) are graphed, what can you conclude about a, b, c, and/or d, if:

a. f(x) and g(x) are parallel?
b. f(x) and g(x) are perpendicular?
c. f(x) does not cross the x-axis?
d. g(x) is vertical?
e. f(x) and g(x) have the same y-intercept?

Problem setup

To do this problem I first had to find out how to graph a linear equation.  I’ll probably have to play around with the variables to see how they graph out on the graphing calculator.


Plans to Solve/Investigate the Problem

I had to have our teacher show me how to interpret the equations, and how to use a graphing calculator.  I learned that in the equation y+mx + b the “m” stands for the slope of a line, which is rise/run, or how high on x axis versus how far on the y axis.  We also reviewed how to create a table on paper of x and y values, and plot them out, then how to use the TI-82 graphing calculator to do the same.  Someone told us that “b” is the y-intercept, or in language I understand, it’s where “x” will be equal to zero. 


Investigation/Exploration of the Problem

In the problem, we use the equations  f(x) = ax + b and g(x) = cx + d. 

a. If f(x) and g(x) are parallel, then I deduced that the slope has to be the same on both.  Therefore a and c will need to be equal.  However b and d do not have to be equal. 

In the following example, the two red lines (k and l) are parallel lines.Parallel lines are written using the symbol | |. In this case k | | l.

b.  If f(x) and g(x) are perpendicular, the slope will change.  By a lot of trial and error on the TI-82, we were able to determine that a and c are negative reciprocals and b & d remained the same. 

Perpendicular Lines:  Two lines are perpendicular if they intersect at a right angle.

  1. If f(x) does not cross the x axis, then it would have a slope of zero.  Therefore, “a” would be 0 and “b” could be anything but zero. 


  1. If  g(x) is vertical, then the slope would be rise/0; therefore the slope would be undefined, because no number can be divided by 0.


  1. If f(x) and g(x) have the same y-intercept, then “b” and “d” would have to be the same number,  and the slopes, “a” and “c” could be anything.  They could even be the same numbers.


Extensions of the Problem

One of the other students talked about how we could use linear equations to solve story problems.  I think I’d like to try that out. 

Author & Contact
Meg Ramsey

Link(s) to resources, references, lesson plans, and/or other materials
Link 1
Link 2


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