I. Simplify.
A. 8 - 3(12x - 4) - 2x = 8 - 36x + 12 - 2x = 20 - 38x
B. (25x + 9) - (55x + 23) = 25x + 9 - 55x - 23 = -30x - 14
II. Solve for x.
A. 5(3x - 2) = 35
15x - 10 = 35
15x - 10 + 10 = 35 + 10
15x = 45
(15x)/15 = 45/15
x = 3
B. 16x - 3(15x + 8) - 6 = 3(5x - 4) - 4(8x - 5)
16x - 45x - 24 - 6 = 15x - 12 - 32x + 20
-29x - 30 = -17x + 8
-29x - 30 + 17x = -17x + 8 + 17x
-12x - 30 = 8
-12x - 30 + 30 = 8 + 30
-12x = 38
(-12x)/(-12) = 38/(-12)
x = -19/6
III. Write each of the following equations in explicit form
(i.e. solve for y) and find three solutions to the equation and then graph.
A. x + 2y = 1 x | y
x + 2y - x = 1 - x 1 | 0
2y = 1 - x 3 | -1
(2y)/2 = (1 - x)/2 5 | -2
y = (1 - x)/2
B. 2x - y = 4 x | y
2x - y - 2x = 4 - 2x -2| -8
-y = 4 - 2x 0 | -4
-y/(-1) = (4 - 2x)/(-1) 3 | 2
y = (4 - 2x)/(-1)
C. y - 3x = 0 x | y
y - 3x + 3x = 0 + 3x 1 | 3
y = 3x 0 | 0
2 | 6
V. Complete the following statements.
A. To find the x-intercept of a line set y = 0 and solve for x.
B. To find the y-intercept of a line set x = 0 and solve for y.
C. Find the x-intercept and the y-intercept of each of the following
and then graph.
1. 4x + 3y = 12 4x - 3y = 12
4x + 3(0) = 12 4(0) - 3y = 12
4x + 0 = 12 0 - 3y = 12
4x = 12 -3y = 12
x = 3 y = -4
Conclusion: The x-intercept is 3 and the y-intercept is -4.
2. 3x - 15 = 15y 3x - 15 = 15y
3x - 15 = 15(0) 3(0) - 15 = 15y
3x - 15 = 0 0 - 15 = 15y
3x = 15 -15 = 15y
x = 5 -1 = y
Conclusion: The x-intercept is 5 and the y-intercept is -1.
VI. Write each of the following lines in slope-intercept form.
Determine the slope and y-intercept of each of the following and then graph.
A. 2y + x = 6 B. 4y - 12 = 0 C. 6x + 3y = 9
2y = -x + 6 4y = 12 3y = -6x + 9
y = (-x + 6)/2 y = 3 y = (-6x + 9)/3
y = -x/2 + 3 y = 0x + 3 y = -2x+ 3
m = -1/2 m = 0 m = -2
b = 3 b = 3 b = 3
VII. Given the following lines. Determine the slope of each line,
the slope of any line parallel to the given line, and the slope
of any line perpendicular to the line.
A. 3x - y = 1
-y = -3x + 1
y = 3x - 1
m = 3
Any line parallel to this line has slope 3.
Any line perpendicular to this line has slope -1/3.
B. 2y + 3x = 6
2y = -3x + 6
y = (-3/2)x + 3
m = -3/2
Any line parallel to this line has slope -3/2.
Any line perpendicular to this line has slope 2/3.
VIII. Find the equation of each of the following lines. Express the equation of the form Ax + By + C = 0.
A. Using the point slope formula we obtain
y - 3 = 5(x - (-2))
y - 3 = 5(x + 2)
y - 3 = 5x + 10
0 = 5x - y + 13
B. Using the slope formula we obtain
m = (10 - 2)/(3 - (-1))
m = 8/4
m = 2
Using the point slope formula we obtain
y - 2 = 2(x - (-1))
y - 2 = 2(x + 1)
y - 2 = 2x + 2
0 = 2x - y + 4
C. Since parallel lines have the same slope we may use the point slope formula to obtain
y - (-1) = 2(x - 4)
y + 1 = 2x - 8
0 = 2x - y - 9
D. Since any line perpendicular to a line with slope 2 has slope -1/2 we may use the point
slope formula to obtain
y - (-1) = -1/2(x - 4)
y + 1 = -1/2x + 2
2y + 2 = -x + 4
x + 2y - 2 = 0
IX. See text for needed formulas.
X. A. Line one is parallel to line two if and only if m = n.
B. Line one is perpendicular to line two if and only if mn = -1.
XI. See answers in back of text book.
I. Simplify.
A. 8 - 3(12x - 4) - 2x = 8 - 36x + 12 - 2x = 20 - 38x
B. (25x + 9) - (55x + 23) = 25x + 9 - 55x - 23 = -30x - 14
II. Solve for x.
A. 5(3x - 2) = 35
15x - 10 = 35
15x - 10 + 10 = 35 + 10
15x = 45
(15x)/15 = 45/15
x = 3
B. 16x - 3(15x + 8) - 6 = 3(5x - 4) - 4(8x - 5)
16x - 45x - 24 - 6 = 15x - 12 - 32x + 20
-29x - 30 = -17x + 8
-29x - 30 + 17x = -17x + 8 + 17x
-12x - 30 = 8
-12x - 30 + 30 = 8 + 30
-12x = 38
(-12x)/(-12) = 38/(-12)
x = -19/6
III. Write each of the following equations in explicit form
(i.e. solve for y) and find three solutions to the equation and then graph.
A. x + 2y = 1 x | y
x + 2y - x = 1 - x 1 | 0
2y = 1 - x 3 | -1
(2y)/2 = (1 - x)/2 5 | -2
y = (1 - x)/2
B. 2x - y = 4 x | y
2x - y - 2x = 4 - 2x -2| -8
-y = 4 - 2x 0 | -4
-y/(-1) = (4 - 2x)/(-1) 3 | 2
y = (4 - 2x)/(-1)
C. y - 3x = 0 x | y
y - 3x + 3x = 0 + 3x 1 | 3
y = 3x 0 | 0
2 | 6
V. Complete the following statements.
A. To find the x-intercept of a line set y = 0 and solve for x.
B. To find the y-intercept of a line set x = 0 and solve for y.
C. Find the x-intercept and the y-intercept of each of the following
and then graph.
1. 4x + 3y = 12 4x - 3y = 12
4x + 3(0) = 12 4(0) - 3y = 12
4x + 0 = 12 0 - 3y = 12
4x = 12 -3y = 12
x = 3 y = -4
Conclusion: The x-intercept is 3 and the y-intercept is -4.
2. 3x - 15 = 15y 3x - 15 = 15y
3x - 15 = 15(0) 3(0) - 15 = 15y
3x - 15 = 0 0 - 15 = 15y
3x = 15 -15 = 15y
x = 5 -1 = y
Conclusion: The x-intercept is 5 and the y-intercept is -1.
VI. Write each of the following lines in slope-intercept form.
Determine the slope and y-intercept of each of the following and then graph.
A. 2y + x = 6 B. 4y - 12 = 0 C. 6x + 3y = 9
2y = -x + 6 4y = 12 3y = -6x + 9
y = (-x + 6)/2 y = 3 y = (-6x + 9)/3
y = -x/2 + 3 y = 0x + 3 y = -2x+ 3
m = -1/2 m = 0 m = -2
b = 3 b = 3 b = 3
VII. Given the following lines. Determine the slope of each line,
the slope of any line parallel to the given line, and the slope
of any line perpendicular to the line.
A. 3x - y = 1
-y = -3x + 1
y = 3x - 1
m = 3
Any line parallel to this line has slope 3.
Any line perpendicular to this line has slope -1/3.
B. 2y + 3x = 6
2y = -3x + 6
y = (-3/2)x + 3
m = -3/2
Any line parallel to this line has slope -3/2.
Any line perpendicular to this line has slope 2/3.
VIII. Find the equation of each of the following lines. Express the equation of the form Ax + By + C = 0.
A. Using the point slope formula we obtain
y - 3 = 5(x - (-2))
y - 3 = 5(x + 2)
y - 3 = 5x + 10
0 = 5x - y + 13
B. Using the slope formula we obtain
m = (10 - 2)/(3 - (-1))
m = 8/4
m = 2
Using the point slope formula we obtain
y - 2 = 2(x - (-1))
y - 2 = 2(x + 1)
y - 2 = 2x + 2
0 = 2x - y + 4
C. Since parallel lines have the same slope we may use the point slope formula to obtain
y - (-1) = 2(x - 4)
y + 1 = 2x - 8
0 = 2x - y - 9
D. Since any line perpendicular to a line with slope 2 has slope -1/2 we may use the point
slope formula to obtain
y - (-1) = -1/2(x - 4)
y + 1 = -1/2x + 2
2y + 2 = -x + 4
x + 2y - 2 = 0
IX. See text for needed formulas.
X. A. Line one is parallel to line two if and only if m = n.
B. Line one is perpendicular to line two if and only if mn = -1.
XI. See answers in back of text book.