Transcript 4.1 Introduction to Linear Equations in Two Variables • A linear equation in two variables can be put in the form (called standard.

4.1 Introduction to Linear Equations in
Two Variables
• A linear equation in two variables can be put in
the form (called standard form):
Ax  By  C
where A, B, and C are real numbers and
A and B are not zero
4.1 Introduction to Linear Equations in
Two Variables
• Table of values (try to pick values such that
the calculation of the other variable is easy):
3x  2 y  12
x
0
y
6
2
3
4
0
4.1 Introduction to Linear Equations in
Two Variables
• Points: (2, 3)
2 is the x-coordinate, 3 is the y-coordinate
• Quadrants:
II x<0 and y>0
I x>0 and y>0
III x<0 and y<0
IV x>0 and y<0
4.2 Graphing by Plotting and
Finding Intercepts
•
•
The graph of any linear equation in two
variables is a straight line.
Note: Two points determine a line.
Graphing a linear equation:
1. Plot 3 or more points (the third point is used
as a check of your calculation)
2. Connect the points with a straight line.
4.2 Graphing by Plotting and
Finding Intercepts
• Graph: 3x  2 y  12
x
0
2
4
y
6
3
0
4.2 Graphing by Plotting and
Finding Intercepts
• Finding the x-intercept (where the line
crosses the x-axis): let y = 0 and solve for x
• Finding the y-intercept (where the line
crosses the y-axis): let x = 0 and solve for y
Note: the intercepts may be used to graph
the line.
4.2 Graphing by Plotting and
Finding Intercepts
• If y = k, then the graph is a horizontal line:
• If x = k, then the graph is a vertical line:
4.2 Graphing by Plotting and
Finding Intercepts
• Example: Graph the equation.
y  3
x
0
2
4
y
-3
-3
-3
4.3 The Slope of a Line
• The slope of a line through points (x1,y1)
and (x2,y2) is given by the formula:
y2  y1
rise
m

x2  x1
run
4.3 The Slope of a Line
• A positive slope rises
from left to right.
• A negative slope falls
from left to right.
4.3 The Slope of a Line
• If the line is
horizontal, m = 0.
• If the line is vertical,
m = undefined.
4.3 The Slope of a Line
•
Finding the slope of a line from its equation
1. Solve the equation for y.
2. The slope is given by the coefficient of x
• Example: Find the slope of the equation.
3x  2 y  5
2 y  3x  5
y   32 x  52  m   32
4.4 The Slope-Intercept Form of a Line
• Standard form:
Ax  By  C
• Slope-intercept form: y  m x  b
(where m = slope and b = y-intercept)
4.4 The Slope-Intercept Form of a Line
• Example: Put the equation 2x + 3y = 6 in slopeintercept form, determine the slope and intercept,
then graph.
2 x  3 y  6  3 y  2 x  6
y  32 x  2  m  32 , b  2
Since b = 2, (0,2) is a point on the line.
Since m   23 , go down 2 and across 3 to point
(3,0) a second point on the line, then connect the
two points to draw the line.
4.4 The Slope-Intercept Form of a Line
• Example: Graph the equation.
2x  3 y  6
x
0
3
y
2
0
4.5 Writing an Equation of a Line
• Standard form: Ax  By  C
Definition is now changed as follows:
A, B, and C must be integers with A > 0
• Slope-intercept form: y  m x  b
(where m = slope and b = y-intercept)
• Point-slope form: y  y1  mx  x1 
for a line with slope m going through point
(x1, y1).
4.5 Writing an Equation of a Line
• Example: Find the equation of a line going
through the point (2,5) with slope = 3. Express
your answer in slope-intercept form.
Start with the point-slope equation:
y  5  3( x  2)
Solve for y to get in slope intercept form:
y  5  3x  6
y  3x  1
4.5 Writing an Equation of a Line
• Example: Find the equation of a line going
through the points (-3,5) and (0,3). Express your
answer in standard form.
Solve for the slope:
m
35
2

0  (3)
3
Use slope intercept form & multiply by the LCD:
y   23 x  3  3 y  2 x  9
2x  3y  9
4.6 Parallel and Perpendicular Lines
•
Parallel lines (lines that do not intersect) have
the same slope.
m1  m2
•
•
Perpendicular lines (lines that intersect to form a
90 angle) have slopes that are negative
reciprocals of each other.
a
b
if m1 
then m2  
b
a
Horizontal lines and vertical lines are
perpendicular to each other
4.6 Parallel and Perpendicular Lines
• Example: Determine if the lines are parallel,
perpendicular or neither: x  3 y  5
y  3x  1
get the slope of each line
x  3y  5  3y  x  5
y   13 x  53  m1   13
y  3x  1  y  3x  1  m2  3
the slopes are negative reciprocals of each other so
the lines are perpendicular
4.6 Parallel and Perpendicular Lines
• Example: Find the equation in slope-intercept
form of a line passing through the point (-4,5) and
perpendicular to the line 2x + 3y = 6
(solve for y to get slope of line)
2 x  3 y  6  3 y  2 x  6
y
2
3
x2m
2
3
(take the negative reciprocal to get the  slope)
m   32   32
4.6 Parallel and Perpendicular Lines
• Example (continued): m  32
Use the point-slope form with this slope and the
point (-4,5)
y  5  32 x  (4)
y  5  32 x  4  32 x  6
In slope intercept form:
y  32 x  11