Finding equations of lines

Section 1-3: Finding Equations of Lines

Linear equations can be written in many different ways. The following chart represents a few of the more useful methods:

The general form	Ax + By = C	Form used for answers.. A,B,C cannot be fractions
The slope-intercept form	y = mx + b	Line has slope m and y-intercept b.
The point-slope form	y-y₁ = m(x-x₁)	Line has slope m and contains (x₁, y₁)
The intercept form		Line has x-intercept a and y-intercept b.

Sample problems

1) Find an equation of the line with x-intercept 6 and y-intercept 3.

Solution: Use the intercept method from above with a = 6 and b = 3.

x/6 + y/3 = 1 multiply by 6 to eliminate the fractions x + 2y = 6

2) Find an equation of the line through (1, 5) and (5, -1).

Solution: Use point-slope form from above and find the slope first.

m = (5 - (-1))/(1 - 5) = 6/-4 = -3/2 Now use the point- slope form using one of the given points. It doesn't matter which one! (y - 5) = (-3/2)(x - 1) multiply by 2 to eliminate the fraction. 2y - 10 = -3x + 3 Put in general form 3x + 2y = 13 (Voila!)

3) Find an equation of the line with y-intercept -2 and parallel to the line 4x + y = 2.

Solution: The equation has the same slope as the given line because it is parallel. Find the slope of this line first.

4x + y = 2 y = -4x + 2 The slope m = -4 Now use the slope-intercept method to find the equation. y = -4x - 2 4x + y = -2

4) Write an equation of the perpendicular bisector of the segment joining (-2,3) and (4, -5).

Solution: We need to do two things. First find the midpoint and then find the slope of the line.

Step 1: Use the midpoint formula to find the midpoint!

This is the point that the perpendicular bisector goes through. Now find the slope. m = (-5 - 3)/(4 - (-2)) = -8/6 = -4/3 But the perpendicular slope is a negative reciprocal. So, m = 3/4 Now use the point-slope form and get: y - (-1) = (3/4)(x - 1) Multiply by 4 to eliminate the fraction. 4y + 4 = 3x - 3 Put in the general form: -3x + 4y = -7 or 3x - 4y = 7 (why?)

This should give you enough to stay ahead of the road-runner!! Beep! Beep! (Is this the Acme web page or what!!)