Math 96--Hand-out on Ideas About Linear Equations--page 1
There are several main ideas about linear equations, graphing, and formulas that it’s useful for you to know.
1.
Ax + By = C is the standard form of a linear equation.
W hen a line is in this form, there are a couple of “ implied” ideas. First, the equation should be in the order with x and y
on the left of the equals sign and the constant on the right. Second, all the numbers should be whole numbers. If they
aren’t, multiply by a common denominator so they will be. Third, the x coefficient should be positive. If it isn’t, multiply
by negative 1 to make the x coefficient positive.
2.
The standard form Ax + By = C is useful for graphing. The easiest way to graph with this form is to find the x-intercept
and the y-intercept. The x-intercept is the point where y = 0 and you’re solving for x. The y-intercept is the point where
x = 0 and you’re solving for y. Once you have the two points, then you can graph.
Graph 5x ! 3y = 30
Let y = 0
5x ! 3(0) = 30
5x
= 30
x
= 6
(6, 0)
So now you have the points (6, 0) and (0, ! 10) and can graph
Let x = 0
5(0) ! 3y = 30
! 3y = 30
y = ! 10
(0, !10)
the line.
3.
x = a number is a vertical line through the x-axis at the point where x = the number
4.
y = a number is a horizontal line through the y-axis at the point where y = the number
5.
6.
7.
8.
When you have both x and y in the equation, you get a “ slanted” line (it isn’t vertical and it isn’t horizontal).
Slope is the steepness of the “ slant” of a line between two points. If you draw the two points and the line between them
and then draw the right triangle that goes with those two points, slope could be said to be the ratio (the comparison) of the
height to the width.
Slope = m=
rise =
run
up/down =
right/left
change in y = y 2 ! y 1
change in x
x2 ! x 1
Given any two points, you can find the slope by using the slope formula.
a.
Find the slope of the line between (! 6, 3) and (! 4, 10).
m = y 2 ! y 1 = 10 ! 3 = 10 ! 3 = 7
x2 ! x1
!4 ! (!6)
!4 + 6
2
So the slope is rise 7 and run 2. From any point on the line, you move 7 upward and 2 to the right to get to another point
on the line.
b.
Find the slope of the line between (7, ! 8) and (3, ! 5).
m = y 2 ! y 1 = !5 ! (!8) =
x2 ! x1
3! 7
!5 + 8
3! 7
=
3
!4
or !3 = ! 3
4
4
So the slope is rise 3 and run !4. From any point on the line, you move 3 upward and 4 to the left to get to another point
on the line. OR the slope is rise !3 and run 4. From any point on the line, you move 3 downward and 4 to the right to
get to another point on the line.
9.
The slope of a line that makes a 45 o angle (from the horizontal) is 1. That means from one point to the next is a rise 1 and
run 1 (go up 1 and right 1 to get from one point to another point).
10.
The slope of lines that make an angle smaller than 45 o (from the horizontal) will be smaller than 1. Why?
11.
The slope of lines that make an angle larger than 45 o (from the horizontal) will be greater than 1. Why?
Math 96--Hand-out on Ideas About Linear Equations--page 2
12.
The slope of a horizontal line is zero. Why? An equation that says y = a number is a horizontal line; therefore, an
equation that says y = a number has a slope of zero.
13.
The slope of a vertical line is undefined. Why? An equation that says x = a number is a vertical line; therefore, an
equation that says x = a number has a slope that is undefined.
14.
Parallel lines have the same slope. If one line has a slope of 6, a parallel line also has a slope of 6.
15.
Perpendicular lines have slopes whose product is negative 1. Perpendicular lines have slopes that are reciprocals of each
other and with opposite signs. If one line has a slope of 2/3, the perpendicular line has a slope of !3/2. If one line has a
slope of !1/4, the perpendicular line has a slope of + 4/1.
16.
y = mx + b is the slope-intercept form of a line.
To graph with this form, place the y-intercept on the line. The y-intercept is where x= 0 so in this equation, you get the
point (0, b). From this point, use the slope to move to another point. You know that slope is rise over run.
Graph y =
2 x+ 5
3
Place (0, 5) on the graph; move up 2 and right 3
Graph y = 5x ! 3
Place (0, !3) on the graph; move up 5 and right 1
Graph y = ! 3 x + 1
5
Place (0, 1) on the graph; move down 3 and right 5 OR
place (0, 1) on the graph; move up 3 and left 5 (notice that the negative goes with the numerator
OR the denominator but not with both)
17.
y ! y 1 = m(x ! x 1) is the point-slope form of a line.
18.
The point-slope form is used to find the equation of a line when you are given the slope and one point, when you are given
two points (because you can find the slope on your own and then use one of the points in the formula), when you are given
a point and the fact that the line is parallel to another line (because the line has the same slope as the parallel line so you
now have slope and a point), and when you are given a point and the fact that the line is perpendicular to another line
(because the line has a negative reciprocal slope to the perpendicular line so you now have the slope and a point). When I
am asked to find the equation of a line, I use the point-slope form.
a.
Find the equation of a line through (5, 8) with a slope of 6.
y ! y 1 = m(x ! x 1)
y ! 8 = 6(x ! 5)
y ! 8 = 6x ! 30
y
= 6x ! 30 + 8
y
= 6x ! 22
Point-slope form
Plug in y 1, x 1, and m
Distribute
Isolate y
Simplify to slope-intercept form
The equation of the line that goes through (5, 8) with a slope of 6 is y = 6x ! 22. Some books (and some instructors) then
require you to rewrite this form into standard form. To do that, you need x and y on the left and the constant on the right.
You also eventually need the x portion to be positive (so if you need to, you divide or multiply the entire equation by !1 to
make the x part positive). Let’s see what happens if we do that.
y = 6x ! 22
!6x + y = ! 22
6x ! y = 22
Equation in slope-intercept form
“ M ove” 6x to the left
M entally divide (multiply) everything by !1 to make x positive
Equation in standard form
So, y = 6x ! 22 and 6x ! y = 22 mean the same thing--they are BOTH correct. In our textbook, we generally will
leave the answer in slope-intercept form so y = 6x ! 22 for this problem.
Math 96--Hand-out on Ideas About Linear Equations--page 3
b.
Find the equation of a line through (!9, 4) and (6, 1).
1! 4 =
6 ! (!9)
First find slope: m =
1!4
6+ 9
=
!3 =
15
!1
5
y ! y 1 = m(x ! x 1)
y ! 4 = ! 1 [ x ! (!9) ]
5
Point-slope form
Plug in y 1, x 1, and m
y ! 4 = ! 1 (x + 9)
5
Simplify the double negative
y!4
= !1 x! 9
5
5
Distribute
y
= ! 1 x! 9+
5
5
y
= ! 1 x! 9 +
5
5
y
= ! 1 x+
5
4
1
20
5
11
5
Isolate y by adding 4; 4 means 4 over 1
Get common denominator
Simplify
Some books present this method (which may be easier for some people):
At the step:
y ! 4 = ! 1 (x + 9)
5
multiply by common denominator 5 to eliminate fraction
5 (y ! 4) = 5 @ ! 1 (x + 9)
5
5y ! 20 = !1(x + 9)
5y ! 20 = !1x ! 9
5y = !1x ! 9 + 20
5y = !1x + 11
y = ! 1 x + 11
5
5
notice 5 cancels the denominator 5!
Now divide all terms by 5 to isolate y
Notice you get the same answer both ways! If the book wanted the answer in standard form, this second procedure would
have been much easier. At the step 5y = !1x + 11, you would add 1x to both sides of the equation to get 1x + 5y = 11
(or simply x + 5y = 11).
So the equation of the line through (!9, 4) and (6, 1) is either
y= ! 1 x+
5
11 (in slope intercept form) or x + 5y = 11 (in standard form).
5
Be sure you know which form the book/instructor prefers!