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M 1313
Review Material
1
Line Formulas:
Point-slope:
y − y1 = m(x − x1 )
m= slope, (x1 , y1 ) is the point
Slope-intercept: y = mx + b
m = slope, b = y intercept
General or Standard form: ax + by = c , where a, b and c are
integers.
Example 1:
Write an equation for a line that has slope
passes through the point (3. − 9 ) .
1
and
5
Example 2: Write an equation for a line which passes through the
points (2 , −2 ) and (− 3 ,2 ) .
Example 3: Find the slope and the y intercept for the equation
2 x − 3y = 9
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Review Material
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Intercepts:
x-intercept means y=0
y-intercept means x=0
Example 4: Find the x and y intercept of the following equation.
1
1
5
x− y=
3
7
8
Example 5: Find the equation of line that has slope of −
y intercept of − 6 .
2
and a
3
Example 6: Find the equation of line that has a slope of − 3 and
x-intercept of − 2 .
Parallel and perpendicular lines:
Parallel lines have the same slope.
Perpendicular lines have slopes that are negative reciprocals.
(This means their product is − 1 ).
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Review Material
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Example 7: Given the point (2 , −1) and the equation 2 x + 3 y = −4
a. Find equation of the line that is parallel at this point.
b. Find equation of the line that is perpendicular at this point.
Solutions to two equations, two unknowns:
One point:
No solution:
All real numbers:
Same Line
Example 8:
Find all solutions:
5x + 2 y = 2
7x + 3y = 6
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Review Material
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Example 9: Find all solution:
a.
b.
2x + y = 5
4x + 2y = 8
2x + y = 4
− 6 x − 3 y = −12
Talk about the possible slopes of lines:
Positive Slope:
Rising right
Falling left
Negative slope:
Rising left
Falling right
Example 10: Graph 5 x − 2 y = 10
y= 2, x=1
Zero or no slope
Undefined slope
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Review Material
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Inequalities:
< or > means the line is dashed -------≤ or ≥ means a solid line
Example 11: Graph 2 x − 5 y < 20
Example 12: Graph the following inequalities and find the solution:
3x + 2 y ≥ 6
x − 3y < 9
Example 13: The solution to the linear inequality − 4 < 5 x + 3 y
will be the half plane
M 1313
Review Material
Example 14: Find the equation of the following graph:
Example 15: Find the equation of the following graph:
Example 16: Find the equations of the following graph:
6
M 1313
Review Material
Example 17:
I.
II.
7
Which of following are true?
 7 3 
The point 
,
 is a point on the graph 10 x − 2 y > 6 .
 10 16 
L1 : − 2 x + 5 y = 5 and L2 : 4 x − 10 y = −4 are perpendicular
lines.
III. A line through the points (− 4 ,2 ) and (− 4 ,7 ) is a horizontal
line.
IV.
(1,−3) is in the solution set
x > 0 and y ≤ 0 .
Example 18: Which of the following are not true?
I.
II.
y = 2 x − 5 has a slope that falls to the right.
A horizontal line has a slope of zero.
III. A line with negative slope falls to the left.