Q. Determine the equation of the line that passes through the points (0, 0) and (3, 9).
A.
y = 3x
B.
y = 2x
C.
y = 3x + 1
D.
y = x + 1
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Solution
The slope m = (9 - 0) / (3 - 0) = 3. The equation is y = 3x.
Correct Answer: A — y = 3x
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Q. Determine the point of intersection of the lines y = 2x + 1 and y = -x + 4.
A.
(1, 3)
B.
(2, 5)
C.
(3, 7)
D.
(4, 9)
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Solution
Setting 2x + 1 = -x + 4 gives 3x = 3, hence x = 1. Substituting back gives y = 3.
Correct Answer: A — (1, 3)
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Q. Determine the x-intercept of the line 4x - 2y + 8 = 0.
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Solution
Setting y = 0 in the equation gives 4x + 8 = 0, thus x = -2.
Correct Answer: B — 2
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Q. Determine the x-intercept of the line 4x - 5y + 20 = 0.
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Solution
Setting y = 0 in the equation gives 4x + 20 = 0, thus x = -5.
Correct Answer: D — -4
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Q. Determine the x-intercept of the line 5x + 2y - 10 = 0.
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Solution
Setting y = 0 in the equation gives 5x - 10 = 0, thus x = 2.
Correct Answer: B — 5
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Q. Determine the x-intercept of the line given by the equation 2x - 3y + 6 = 0.
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Solution
Set y = 0 in the equation: 2x + 6 = 0 => x = -3.
Correct Answer: B — 3
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Q. Find the angle between the lines y = 2x + 1 and y = -0.5x + 3.
A.
60 degrees
B.
45 degrees
C.
90 degrees
D.
30 degrees
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Solution
The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.
Correct Answer: A — 60 degrees
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Q. Find the distance from the point (1, 2) to the line 3x + 4y - 12 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |3(1) + 4(2) - 12| / sqrt(3^2 + 4^2) = |3 + 8 - 12| / 5 = 1.
Correct Answer: A — 2
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Q. Find the distance from the point (3, 4) to the line 2x + 3y - 6 = 0.
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Solution
Distance = |Ax1 + By1 + C| / sqrt(A^2 + B^2) = |2*3 + 3*4 - 6| / sqrt(2^2 + 3^2) = |6 + 12 - 6| / sqrt(13) = 12 / sqrt(13).
Correct Answer: B — 3
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Q. Find the equation of the line passing through the points (1, 2) and (3, 4).
A.
y = x + 1
B.
y = 2x
C.
y = x + 3
D.
y = 2x - 1
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Solution
The slope m = (4-2)/(3-1) = 1. Using point-slope form: y - 2 = 1(x - 1) gives y = x + 1.
Correct Answer: A — y = x + 1
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Q. Find the equation of the line that is perpendicular to y = 5x + 2 and passes through (2, 3).
A.
y = -1/5x + 4
B.
y = 5x - 7
C.
y = -5x + 13
D.
y = 1/5x + 2
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Solution
The slope of the perpendicular line is -1/5. Using point-slope form: y - 3 = -1/5(x - 2) gives y = -1/5x + 13/5.
Correct Answer: C — y = -5x + 13
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Q. Find the equation of the line that is perpendicular to y = 5x + 2 and passes through the origin.
A.
y = -1/5x
B.
y = 5x
C.
y = -5x
D.
y = 1/5x
Show solution
Solution
The slope of the given line is 5. The slope of the perpendicular line is -1/5. Using y = mx + c, we get y = -1/5x.
Correct Answer: C — y = -5x
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Q. Find the equation of the line that passes through the origin and has a slope of -2.
A.
y = -2x
B.
y = 2x
C.
y = -x
D.
y = x
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Solution
Using the slope-intercept form: y = mx + b, where b = 0, we have y = -2x.
Correct Answer: A — y = -2x
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Q. Find the midpoint of the line segment joining the points (2, 3) and (4, 7).
A.
(3, 5)
B.
(2, 5)
C.
(4, 3)
D.
(5, 6)
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Solution
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 4)/2, (3 + 7)/2) = (3, 5).
Correct Answer: A — (3, 5)
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Q. Find the point of intersection of the lines y = 2x + 1 and y = -x + 4.
A.
(1, 3)
B.
(2, 5)
C.
(3, 7)
D.
(4, 9)
Show solution
Solution
Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).
Correct Answer: A — (1, 3)
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Q. Find the point of intersection of the lines y = x + 1 and y = -x + 5.
A.
(2, 3)
B.
(3, 2)
C.
(1, 2)
D.
(0, 1)
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Solution
Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).
Correct Answer: A — (2, 3)
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Q. Find the slope of the line passing through the points (2, 3) and (4, 7).
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Solution
The slope m is given by (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4 / 2 = 2.
Correct Answer: A — 2
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Q. Find the slope of the line that passes through the points (0, 0) and (5, 5).
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Solution
The slope m = (5 - 0) / (5 - 0) = 1.
Correct Answer: B — 1
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Q. Find the y-intercept of the line represented by the equation 5x - 2y = 10.
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Solution
Set x = 0: -2y = 10 => y = -5. The y-intercept is (0, -5).
Correct Answer: B — 2
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Q. If a line passes through the points (1, 1) and (2, 3), what is its equation in slope-intercept form?
A.
y = 2x - 1
B.
y = 3x - 2
C.
y = 2x + 1
D.
y = x + 2
Show solution
Solution
Slope m = (3 - 1) / (2 - 1) = 2. Using point-slope form: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. If the line 3x + 4y = 12 intersects the x-axis, what is the point of intersection?
A.
(4, 0)
B.
(0, 3)
C.
(0, 4)
D.
(3, 0)
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Solution
Set y = 0 in the equation: 3x = 12 => x = 4. The point is (4, 0).
Correct Answer: A — (4, 0)
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Q. If the line 3x + 4y = 12 is transformed to slope-intercept form, what is the slope?
A.
-3/4
B.
3/4
C.
4/3
D.
-4/3
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Solution
Rearranging gives y = -3/4x + 3. The slope is -3/4.
Correct Answer: A — -3/4
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Q. If the line 3x + 4y = 12 is transformed to slope-intercept form, what is the y-intercept?
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Solution
Rearranging gives y = -3/4x + 3. The y-intercept is 3.
Correct Answer: B — 4
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Q. If the line 3x - 4y + 12 = 0 is parallel to another line, what is the slope of that line?
A.
3/4
B.
-3/4
C.
4/3
D.
-4/3
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Solution
The slope of the line is given by -A/B = -3/-4 = 3/4. Parallel lines have the same slope.
Correct Answer: B — -3/4
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Q. If the line 3x - 4y + 12 = 0 is transformed to slope-intercept form, what is the slope?
A.
3/4
B.
-3/4
C.
4/3
D.
-4/3
Show solution
Solution
Rearranging gives y = (3/4)x + 3, so the slope is -3/4.
Correct Answer: B — -3/4
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Q. If the line 5x + 12y = 60 is transformed to slope-intercept form, what is the slope?
A.
-5/12
B.
5/12
C.
12/5
D.
-12/5
Show solution
Solution
Rearranging gives y = -5/12 x + 5, so the slope is -5/12.
Correct Answer: A — -5/12
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Q. If the line 5x + 2y = 10 intersects the y-axis, what is the y-coordinate of the intersection point?
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Solution
Setting x = 0 in the equation gives 2y = 10, hence y = 5.
Correct Answer: B — 2
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Q. If the line 5x - 2y + 10 = 0 is reflected about the x-axis, what is the new equation?
A.
5x + 2y + 10 = 0
B.
5x - 2y - 10 = 0
C.
5x + 2y - 10 = 0
D.
5x - 2y + 10 = 0
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Solution
Reflecting about the x-axis changes the sign of y-coefficient: 5x + 2y + 10 = 0.
Correct Answer: A — 5x + 2y + 10 = 0
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Q. What is the angle between the lines 2x + 3y - 6 = 0 and 4x - y + 1 = 0?
A.
45 degrees
B.
60 degrees
C.
90 degrees
D.
30 degrees
Show solution
Solution
The slopes of the lines are -2/3 and 4. The angle θ can be found using tan(θ) = |(m1 - m2) / (1 + m1*m2)|.
Correct Answer: C — 90 degrees
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Q. What is the angle between the lines y = 2x + 1 and y = -0.5x + 3?
A.
90 degrees
B.
60 degrees
C.
45 degrees
D.
30 degrees
Show solution
Solution
The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.
Correct Answer: A — 90 degrees
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