Expert: Jeff White - 10/14/2004

Question
From: Massachusetts
Reason for seeking help.  Trying to help my daughter but am finding various aspects of algebra to be very difficult.  Can you help?

Determine the equation of each line (write in slope- intercept form.

The line through (-1, 3) with a slope 1/3.

The line through (3, 5) that is parallel to the x axis.

The line through (0,6) that is parallel to the line 3x + 5y = 15.

The line through (8,0) that is perpendicular to the line x + y = 3.

I shall appreciate any help you can give me.

Sincerely,

Mary Ellen Siener

Answer
What you'll need on all of these is the point-slope formula, which is an equation used when you know the slope of a line and a point that it passes through.  It is:

y - y1 = m(x - x1) (the 1's are subscripts, not exponents)

The purpose of the subscripts is to indicate that the formula is referring to specific numbers, not variables, specifically the point (x1,y1).

Look at the first problem.  It goes through the point (-1,3) and has a slope of 1/3.  So fitting this information to the point-slope formula gives:

y - 3 = 1/3(x - (-1))
y - 3 = 1/3(x + 1)
y - 3 = x/3 + 1/3
y = x/3 + 1/3 + 3
y = x/3 + 1/3 + 9/3
y = x/3 + 10/3

This equation is in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

Problem #2:
A line that is parallel to the x axis has a slope of zero.  You could use the slope-intercept formula, but it's not necessary.  A horizontal line will have an equation of the form y = k, where k is y value of every ordered pair the line passes through.  Since the line goes through (3,5), the equation would be y = 5.

Problem #3:
Lines that are parallel have the same slope.  So you first need to get the slope of the line your line is parallel to.  You can do that by getting it in slope-intercept form:
3x + 5y = 15
5y = -3x + 15
y = (-3/5)x + 3

So its slope is -3/5.  You then need to use that in the point-slope formula:

y - 6 = -3/5(x - 0)
y - 6 = (-3/5)x
y = (-3/5)x + 6

Problem #4:
If two lines are perpendicular, their slopes are negative reciprocals.  So find the slope of the line given:
x + y = 3
y = -x + 3

So the slope is -1.  The negative reciprocal of -1 is 1.  So plug that into the point-slope formula:

y - 0 = 1(x - 8)
y = x - 8

Hope this helps.

Jeff