PART 1 : LECTURE
Linear equation in two variables
Linear equations can be placed in this form y = mx + b m and b are constants. x and y are variables.
which ones of these equations are linear equations?
y = 2x + 1
y = x2 - 3
y = 3 / (x - 4)
4x + y = 7
Show that 4x + y = 7 is a linear equation
They are also called 1st degree equations because the exponent on the variables (x and y) are simply one. examples:
y = 2x + 1
y = x -4
y = -3/4x
All
linear equations have graphs that are straight lines. So it does not
take many points to see the graph. (usually 3 points are enough).
example1: y = 3x - 2
build a x , y table (3 points), plot them and graph the line.
(the points are said to be colinear)
example2:
3x + 2y = 4
First put in the form y = mx + b (isolate y)
then built the table, then trace the line.
example3: (degenerate form) y = 3 (trace)
example4: x = -2 (trace)
example5: 2x - 5y = 10. isolate y then trace the line
example6: y = -2 and x = 3 (trace)
example7: A car is traveling at a constant speed of 40mph.
The distance d, the car travels in t hours is given by d = 40t .
graph this equation for 0<= t <= 5.
First build a (t,d) table then plot distance versus time.
intercepts and slopes of straight lines
x-intercept is the place where the graph crosses the x-axis (y=0)
y-intercept is the place where the graph crosses the y-axis. (x=0)
we can use the intercepts to graph the line.
example1: 2x + 3y = 6
find the x-intercept and the y-intercept.
trace the line
example2: y = 2x - 4
find the x-intercept and the y-intercept. sketch the graph.
The slope gives the steepness of a line. we can attach a number to the steepness.
slope = rise / run
vertical line : the slope is undefined
horizontal line: the slope is zero
slope = change in y / change in x = (y2 - y1) / (x2 - x1)
example3: Find the slope of the line containing the points (-1, 1) and (2,3)
example4: Find the slope of the line passing through (-1,3) and (4,3)
example5: Find the slope of the line passing through (2,-2) and (2,4)
example6: If 2 lines are // they must have the same _____________
example7: suppose we want to graph y = 2x - 3
trace the line. Find the slope and the y-intercept. conclusion?
this
form (y=mx + b) is called the slope-intercept form. If we have
the slope and the y-intercept, it is easy to trace the line.
example8: use this information to trace y = 2/3 x + 1 (start by potting the y-intercept).
example9: 3x + 2y = 12. Find the slope-intercept form and trace.
equation of straight line
We can do the opposite. given the graph, we can find the equation.
example1: find the equation of the line of slope -1/2 and y-intercept (0,3)
It is not always that easy. Sometines, you need to use m = (y2-y1) / (x2 - x1)
or m/1 = (y-y1) / (x - x1)
or y - y1 = m (x - x1) this is called the lope-point form
you can use this form if you are given a point on a graph and the slope.
example2: Find the equation of the line passes through (2, 3 ) and has a slope of -2.
example3: Find the equation of the line passing through (-2, -1) and has a slope of 3/2
example4 Find the equation of the line passing through (-3, -1) and (3,3)
first use the slope formula to find m and then use the slope-point form
example5: What is Hooke's law?
PART II: ON YOUR OWN
from book p.163
1) graph y = 2x
2) graph y = - 1/2 x + 3 (you can use the y-intercept and the slope)
3) graph 3x + 5y =10
from book p. 194
4) graph y = 2x and y = 2x + 3 using the same set of axes and compare the graphs
5) graph y = 1/3x and y = 1/3 x - 2 using the same set of axes and compare the graphs
6) find the y-intercept point y = -5x + 4
7) Find the equation of the line passing through the points ( -4, 3) and (2, - 5). First find the
slop and use y - y1 = m (x- x1). graph the line
8) find the slope and y-intercepts of y = 5x - 4
9) Find the y-intercept and the slope of 2x + 3y = 8
10)
Sales at wharehouse clubs and superstores have increased steadily in
recent years. In 1998, sales of $98.6 billion were registered at these
types of stores. by 2003, sales had reisen to $217.5 billion. Find the
rate of sales with respect to time, in years
from book p.205
11) Find the intercepts of 3x + 2y = 12. then graph the line
12) graph y = - 2/3 x + 1
13) graph y = 3
14) graph x = -2
15) find the equation of the line containing the points (2,3) and (-6, 1)
PART III
quiz from last assignment
PART IV
p. 184 33, 36, 40
p. 203 10, 11, 14, 15, 16,18, 20, 22, 23,27,29
p. 214 3, 7, 11,15, 19, 23, 24, 29, 31, 42, 43
p. 229 10, 15, 22, 30
p. 230 45
PART III - QUIZ
from assignments due today
PART IV - assignments. (next quiz from these problems)
]p.148 13, 15, 30, 32, 35
p. 149 50, 51, 52
p. 150 82, 84, 102, 104
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