4.1 introduction to linear equations in two variables a linear equation in two variables can be put...
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4.1 Introduction to Linear Equations in Two Variables
• A linear equation in two variables can be put in the form (called standard form):
where A, B, and C are real numbers andA and B are not zero
CByAx
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):
1223 yx
x y
0 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:I x>0 and y>0II 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: 1223 yxx y
0 6
2 3
4 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.
3y
x y
0 -3
2 -3
4 -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:
run
rise
12
12
xx
yym
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.
23
25
23
532
523
mxy
xy
yx
4.4 The Slope-Intercept Form of a Line
• Standard form:
• Slope-intercept form: (where m = slope and b = y-intercept)
CByAx
bmxy
4.4 The Slope-Intercept Form of a Line
• Example: Put the equation 2x + 3y = 6 in slope-intercept form, determine the slope and intercept, then graph.
Since b = 2, (0,2) is a point on the line. Since , 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.
2,2
623632
32
32
bmxy
xyyx
32m
4.4 The Slope-Intercept Form of a Line
• Example: Graph the equation.
632 yx
x y
0 2
3 0
4.5 Writing an Equation of a Line
• Standard form: Definition is now changed as follows:A, B, and C must be integers with A > 0
• Slope-intercept form: (where m = slope and b = y-intercept)
• Point-slope form: for a line with slope m going through point (x1, y1).
11 xxmyy
bmxy
CByAx
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:
Solve for y to get in slope intercept form:
13
635
)2(35
xy
xy
xy
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:
Use slope intercept form & multiply by the LCD:
932
9233
3
2
)3(0
53
32
yx
xyxy
m
4.6 Parallel and Perpendicular Lines
• Parallel lines (lines that do not intersect) have the same slope.
• Perpendicular lines (lines that intersect to form a 90 angle) have slopes that are negative reciprocals of each other.
• Horizontal lines and vertical lines are perpendicular to each other
21 mm
a
bmthen
b
amif 21
4.6 Parallel and Perpendicular Lines
• Example: Determine if the lines are parallel, perpendicular or neither:
get the slope of each line
the slopes are negative reciprocals of each other so the lines are perpendicular
13
53
xy
yx
31313
5353
2
31
135
31
mxyxy
mxy
xyyx
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)
(take the negative reciprocal to get the slope)
32
32 2
623632
mxy
xyyx
23
23 m
4.6 Parallel and Perpendicular Lines
• Example (continued):Use the point-slope form with this slope and the point (-4,5)
In slope intercept form:
11
645
)4(5
23
23
23
23
xy
xxy
xy
23m