Download Angles, Degrees, and Special Triangles

Linear Equations in Two Variables
MATH 109 - Precalculus
S. Rook
Overview
• Section 1.3 in the textbook:
– Graphing a linear equation by using its slope
– Finding the slope of a line
– Writing Linear Equations
– Parallel and Perpendicular Lines
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Graphing a Linear Equation by
Using its Slope
Graphing Lines in Slope-Intercept Form
• Slope-Intercept Form of a line: y = mx + b
where m is the slope and the y-intercept is
(0, b)
• To graph a line using its slope and y-intercept:
– Solve the equation for y if necessary
– Plot the y-intercept (0, b)
– Use rise over run with the slope to get 2 or 3
more points
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Other Lines
• Two other type of lines:
– When a linear equation is only in 1 variable (x or y)
– Vertical lines have the form x = a (a is a constant):
• m = Ø (undefined)
• To sketch, find x = a on the x-axis and draw a vertical
line
– Horizontal lines have the form y = b (b is a constant)
• m=0
• To sketch, find y = b on the y-axis and draw a horizontal
line
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Graphing a Linear Equation by
Using its Slope (Example)
Ex 1: Sketch the graph of the linear equation
using its slope and y-intercept:
a) x  y  5
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b) y  x  3
3
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Graphing Other Lines (Example)
Ex 2: Sketch the line:
a) y + 3 = 2
b) x = 4
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Finding the Slope of a Line
Definition and Properties of Slope
• Slope (m): the ratio of the change in y (Δ y) and the
change in x (Δ x)
– Quantifies (puts a numerical value on) the “steepness” of a
line
• Given 2 points on a line, we can find its slope:
y y2  y1
m

x x2  x1
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Sign of the Slope of a Line
• To determine the sign of the slope, examine
the graph of the line from left to right:
– Positive if the line rises
– Negative if the line drops
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Finding the Slope of a Line
(Example)
Ex 3: Find the slope of the line:
a) Through (-2, 5) and (4, -5)
b) Vertical line through (3, 11)
c) Through (0, 8) and (1, 10)
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Writing Linear Equations
Point Slope Formula & Standard Form
of a Line
• Given the slope and a point on a line, we can
construct its equation
• Point-slope formula: y – y1 = m(x – x1)
–
–
–
–
(x1, y1) is any point
x and y are variables
Very similar to the slope formula
Could also use y = mx + b
• Standard form: a linear equation in the form
Ax + By = C where A, B, and C are constants
– Variables to the left and constants on the right
– NO fractions or decimals
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Writing an Equation in Standard
Form
Ex 4: Write the linear equation of the line in
standard form:
a) Through (-4, -8) with a slope of -¼
b) Through (2, 0) with a slope of 1
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Equations of Lines when Given Two
Points
• We know how to find the slope of a line given
two points
• Proceed as before
– Pick one of the two points to use in the pointslope formula
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Writing the Equation of a Line
Given Two Points (Example)
Ex 5: Write the equation of the line in slopeintercept form:
a) Through (4, 7) and (8, 9)
b) Through (2, 1) and (-3, 6)
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Equations of Vertical and Horizontal
Lines
• Recall the slopes of vertical and horizontal
lines:
– Slope of a vertical line is undefined
– Slope of a horizontal line is zero
• Use the slope along with the given point to
construct the equation of the line:
– If the line is vertical, use the x-coordinate of the
given point
– If the line is horizontal, use the y-coordinate of
the given point
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Equations of Vertical and
Horizontal Lines (Example)
Ex 6: Write the equation of the line:
a) Through (-2, 5) with undefined slope
b) Through (4, 13) with a slope of zero
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Parallel and Perpendicular Lines
Slopes of Parallel and Perpendicular
Lines
• Parallel lines: two lines that have the SAME
slope
• Perpendicular lines: two lines that have
OPPOSITE RECIPROCAL slopes
– i.e. the product of the slopes is -1
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Classifying Lines as Parallel,
Perpendicular, or Neither (Example)
Ex 7: Determine whether the two lines are
parallel, perpendicular, or neither:
a) 3x + y = -4 and 6y – 2x = 12
b) y = 2x + 1 and y = -2x – 3
c) 3y – 15x = -6 and y = 5x + 2
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Equations of Parallel and
Perpendicular Lines
• Given the equation of a line, we want to find the
equation of a second line that is parallel or perpendicular
to the first
• Slope is not explicitly given
– Put the equation of the given line in slope-intercept form
• Determine the appropriate slope based on whether the
second line is to be parallel or perpendicular to the first
• Use the slope and the given point in the point-slope
formula or y = mx + b
• A vertical line is parallel to another vertical line and a
horizontal line is perpendicular to a vertical line
– Vice versa for a horizontal line
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Equations of Parallel and
Perpendicular Lines (Example)
Ex 8: Write the equation of the line in standard
form if possible:
a) Perpendicular to y = -3 and passes through (1, -5)
b) Parallel to y = ½x + 5 and passes through (4, 2)
c) Perpendicular to 3x + y = -1 and passes through
(-1, -1)
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Summary
• After studying these slides, you should be able to:
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–
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Graph a line using its slope and y-intercept
Graph vertical or horizontal lines
Find the slope of a line
Write linear equations
Classify two lines as being parallel, perpendicular, or
neither
– Write equations involving parallel and perpendicular lines
• Additional Practice
– See the list of suggested problems for 1.3
• Next lesson
– Functions (Section 1.4)
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