Graphing Linear Equations
Cartesian Coordinate System
Quadrants – There are four quadrants in the Cartesian coordinate
system created by the x-axis (the horizontal axis) and the yaxis (the vertical axis.)
Ordered pair (x,y) ORDER IS IMPORTANT – The first number
listed must be the value of x, and the second number listed
must be the value of y. The ordered pair (3,5) is graphed to
the left. Note that the lines from the point connect to the 3 on
the x--axis, and the 5 on the y-axis.
Origin – ordered pair of (0,0) located at the intersection of the x
and y-axes. See the origin labeled on the above graph.
x-intercept – ordered pair where the line crosses the x-axis. (x,0) Note that y = 0
y-intercept - ordered pair where the line crosses the y-axis. (0, y) Note that x = 0
Slope of a line: For 2 ordered pairs  x1 , y1  and  x2 , y2 
slope of a line =
y  y1
change in y (vertical change or rise)
 2
change in x (horizontal change or run) x2  x1
Ex. For the points: (  3,  6) and (0,  4) the slope is
 4  ( 6) 2

0  ( 3)
3
1. Slope Summary:
a. If the slope is positive, the line will rise going from left to right.
b. If the slope is negative, the line will fall going from left to right.
c. The slope of a horizontal line (a line of the form: y = a number) is 0.
d. The slope of a vertical line (a line of the form: x = a number) is undefined.
e. If the slopes of two lines are equal, then they are parallel.
f. If the slopes of two lines are the negative reciprocals of each other, then they are perpendicular.
(the product of the slopes will be  1)
2. Forms of linear equations:
a. Standard form
b. Slope-Intercept form
ax+by=c
y= mx+b where m is the slope, and (0, b) is the y-intercept of the line
c. Point-Slope form
y  y1  m( x  x1 ) where m is the slope, and ( x1 , y1 ) is a point on the line.
3. Graph Linear Equations by:
a.
b.
c.
d.
Plotting points
Using the x and y intercepts
Using a point and the slope
Using y = mx + b (slope-intercept form of the equation)
4. Finding Equations to a line:
a. Given: Slope and the y-intercept, use the slope-intercept formula, y = mx + b.
Ex. Given m = 8 and y-intercept = (0,  3), use substitution into the formula and get y = 8x  3
b. Given: One point and the slope , use either the slope-intercept formula or the point-slope formula.
Ex. Given m = 6 and the point (  5, 1)
A. Using the slope-intercept formula, substitute in the values of x, y, and m and solve for b.
y = mx + b
1 = 6(  5) + b
1 =  30 + b
+30 +30
31 = b
Now substitute the values of x and b in the slope-intercept form of the equation
y = 6x + 31
B. Using the point-slope formula, substitute in the values of.
x1 , y1 , and m and then solve the equation for y
y  y1  m( x  x1 )
y  1 = 6(x  (  5))
y  1 = 6(x + 5)
y  1 = 6x + 30
+1
+1
y = 6x + 31
c. Given: Two points, use the slope intercept-formula or the point-slope formula.
Ex. Given (0,3) and (  1,7)
First calculate the slope: m =
73
4

 4
1  0 1
Next choose one of the points and proceed to find the equation of the line using the directions given
above in 4.b. when given one point and the slope. You can use either the slope-intercept or pointslope formula.
A.
Using the slope-intercept formula, substitute in the values of x, y, and m and solve for b.
y = mx + b Choose one of the points given; ex. (0, 3) to use for the x & y values
3 =  4(0) + b
3=b
Now substitute the values of m and b in the slope-intercept form of the equation
y =  4x + 3
B. Using the point-slope formula, substitute in the values of
x1 , y1 , and m and then solve the equation for y
y  y1  m( x  x1 ) Choose one of the points given; ex. (0,3) to use for the x & y values
y  3   4( x  0)
y  3   4x
y   4x  3