Math 1313
Prerequisites
Equations (These will not be given on the test. You need to memorize them.)
y 2 − y1
x 2 − x1
Slope-Intercept Form: y = mx + b
Point-Slope Form: y − y1 = m( x − x1 )
Standard Form: ax + by = c
Slope of a line: m =
Example 1: Write an equation for the line that has slope
1
and passes through (3, -9).
5
Example 2: Write an equation for the line that passes through (-2, 5) and (4, 8).
Math 1313 - Prerequisites
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Parallel and Perpendicular Lines
Two lines are parallel if and only if their slopes are the same.
Two lines are perpendicular if and only if their slopes are negative reciprocals of each
other.
Example 3: Find an equation of the line that passes through the point (-2, 2) and is
parallel to the line 2x – 4y – 8 = 0.
Example 4: Find an equation of the line that passes through the point (-1, -3) and is
perpendicular to the line that passes through (3, -4) and (9, -6).
Example 5: Find the point of intersection.
2x − 5 y = 2
a.
x − 4 y = −2
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b. 2 y = 4 − x
3x + 6 y = 13
c. 2 x − y = 3
2 y = 4x − 6
Example 6: Graph the following system of inequalities.
x− y ≥6
a.
x + y ≤1
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3x + 2 y > 6
b. x − 3 y < 9
x≥0
Example 7: Write a system of linear inequalities that describes the shaded region.
a.
y
3
2
1
x
-4
-3
-2
-1
1
2
3
4
5
-1
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b.
y
3
2
1
x
-3
-2
-1
1
2
3
4
5
6
-1
c. The slope of the solid line with positive slope is 1. The slope of the dashed line is
2
− .
5
y
3
2
1
x
−3
−2
−1
1
2
3
4
5
6
−1
Math 1313 - Prerequisites
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Example 8: Which of the following are true?
⎛7 3⎞
I. The point ⎜ , ⎟ is on the graph of 10 x − 2 y > 6.
⎝ 10 16 ⎠
II. The following two lines are perpendicular to each other.
Line L1 : − 2 x + 5 y = 5
⎛6 ⎞
Line L2 : passes through (4, 6) and ⎜ ,2 ⎟.
⎝5 ⎠
III. A line through (3, 4) and (3, 5) is a horizontal line.
IV. The line y = 2 x − 2 rises to the left.
V. The point (1, -3) is a point in the solution set of x > 0 and y ≤ 0 .
Example 9: Which of the following are false?
I. The slope of the line through (3, -2) and (5, -2) is undefined.
⎛5 ⎞
II. The point ⎜ ,0 ⎟ is on the graph of 4 x − 7 y > 5.
⎝4 ⎠
III. Line L1 has slope m. Line L2 is parallel to line L1 , so L2 has slope − m .
IV. The point (1, 2) is in the solution set of y > − x − 1 .
V. The solution to − 2 x − 3 y ≥ 4 is the half-plane lying below the line − 2 x − 3 y = 4 .
Math 1313 - Prerequisites
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