   SEARCH HOME Math Central Quandaries & Queries  Question from Lisa, a student: I'm having a hard time understanding the concept of how to write an equation of the line containing the given points and parallel to the given line. Express the answer in the form of y = mx + b. (7,9); 2x+5y=4. i would appreciate it alot if someone can show ne how to do this. Thank you, Lisa Hi Lisa,

If you have the equation of a line in the form y = mx + b then you know immediately that the slope of the line is m. Similarly if you know that a line has slope m then you know that there is a number b so that the equation of the line can be written y = mx + b.

I'm going to solve a problem that is similar to yours.

Write an equation of the line containing the given points and parallel to the given line.
Express the answer in the form of y = mx + b.
(2, 3); 3x + 4y = -5.

I am going to first write the equation 3x + 4y = -5 in the form y = mx + b since that will tell me the slope.

3x + 4y = -5

4y = -3x - 5

divide both sides by 4

y = -3/4 x - 5/4.

This is in the required form y = mx + b so I know that m = -3/4 is the slope of the given line.

Parallel lines have the same slope so the line I am seeking also has slope m = -3/4. Thus there is some number b so that the line I want is y = -3/4 x + b. All that remains is to find b.

The extra fact I know is that the point (2, 3) is on this line, so substituting x = 2, y = 3 I get

3 = -3/4 (2) + b

and hence

b = 3 + 3/2 = 9/2.

Hence the equation of the line parallel to 3x + 4y = -5 and containing (2, 3) is

y = -3/4 x + 9/2.     