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GCSE Straight Line Equations
Exercise 1 β Lines and their Equations
Question 1: Draw the line with equation π₯ + π¦ = 2 (Note the scales)
Question 2: Draw the line with equation π¦ = β1
2π₯ + 1
Question 3: When the point (3, π) lies on each of these lines, find the value of π. a) π¦ = 3π₯ + 2 b) π¦ = 4π₯ β 2 c) π¦ = 3 β 2π₯ d) π₯ + π¦ = 7 e) π₯ β 2π¦ = 1 Question 4: Give the coordinate of the point where each line crosses the (a) π¦-axis and (b) the π₯-axis.
i) π¦ = 3π₯ + 1 ii) π¦ = 4π₯ β 2
iii) π¦ =1
2π₯ β 1
iv) 2π₯ + 3π¦ = 4 Question 5: On the grid, draw the graph of y = 4x β 2 (note the axis scales)
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Question 6: When the point (π, 3) lies on each of these lines, find the value of π. a) π¦ = 2π₯ + 1 b) π¦ = 2π₯ β 1 c) π¦ = 8 β 2π₯ d) 2π₯ + 3π¦ = 4 Question 7. (a) Complete the table of values for 3x + 2y = 6
x β2 β1 0 1 2 3
y 4.5 3 β1.5
(b) On the grid, draw the graph of 3x + 2y = 6
Question 8: Complete the table of values for π₯ + 2π¦ = 1.
π β2 β1 0 1 2
π 1
Question 9: Put a tick or cross to determine whether each of the following points
are on the line with the given equation.
π¦ = 1 β π₯ π₯ + 2π¦ = 3
(3, β2)
(1,2)
(2,1
2)
(β1,2)
Question 10: For the given equation of a line and point, indicate whether the
point is above the line, on the line or below the line. (Hint: Find out first what π¦ is
on the line for the given π₯)
Below the line On the line Above the line
π¦ = 3π₯ + 4 (3,11)
π₯ + π¦ = 5 (7, β2)
π¦ = 3 β 2π₯ (β3,10)
2π₯ + 3π¦ = 4 (3
4,4
5)
Question 1: The equation of a line is ππ₯ + ππ¦ = π. If the π₯ value of some point
on the line is π, what is the full coordinate of the point, in terms of π, π, π, π?
Question 2: What is the area of the region enclosed between the line with
equation 2π₯ + 7π¦ = 3, the π₯ axis, and the π¦ axis?
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Exercise 2: Gradients and Midpoints
1. By rearranging the equations into the form π¦ = ππ₯ + π, determine the
gradient of each line.
a. π¦ = π₯ + 1
b. π¦ = 2 β π₯
c. π¦ = 3
d. 2π¦ = 6π₯ β 4
e. 4π¦ = 5π₯ + 1
f. π₯ + π¦ = 1
g. 2π₯ + 3π¦ = β4
h. π₯ β 3π¦ = 4
i. π₯ + 4π¦ = 5
j. 3π₯ β 4π¦ = 7
2. Determine the gradient of the line which goes through the following
points.
a. (0,0), (2,2)
b. (1,3), (3,7)
c. (0,5), (4,25)
d. (2,2), (β1,5)
e. (4,3), (10,6)
f. (7,8), (β4,β3)
g. (7,1), (β1,5)
h. (6,5), (8,1)
i. (1,3), (5,10)
j. (β1,4), (9, β5)
k. (1,0), (β2,β4)
3. Determine the midpoint of π΄ and π΅.
a. π΄(3,6), π΅(5,8)
b. π΄(3,6), π΅(19, 9)
c. π΄(3,6), π΅(β1,β6)
d. π΄(β1, 5, 4), π΅(β7,β1, 9)
4. If ππ₯ + ππ¦ = 1, where π and π are constants, determine the gradient of
the line in terms of π and π.
5. If π is the midpoint of π΄π΅, and π΄ = (4,β3),π = (1,1), what is the
coordinate of π΅?
6. If π΄(4,4), π΅(16,34) and πΆ is a point on the line π΄π΅. Find the coordinates
of πΆ when:
a. π΄πΆ: πΆπ΅ = 1: 3
b. π΄πΆ: πΆπ΅ = 2: 3
7. A triangle π΄π΅πΆ has the coordinates π΄(0,3), π΅(6,8), πΆ(β1,3). A new
triangle is formed by joining the midpoints of each of the sides.
Determine the gradients between each of the three points.
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Exercise 3: Equations given gradient/points
1. Find the equation of the line with the specified gradient which goes
through the specified point, leaving your answer in the form π¦ = ππ₯
a. (4,3), π = 2
b. (5,20), π = 3
c. (4,0), π = 5
d. (4,3), π =1
2
e. (β4,3), π = β1
f. (6,4), π = β1
3
2. Do the same as above, but leave your equations in the form ππ₯ + ππ¦ +
π = 0 where π, π, π are integers. (I advise using the formula)
a. (2,3), π = 4
b. (5,11), π =1
2
c. (7, β2),π =1
3
d. (β2,5),π =2
3
e. (4, β1),π =3
4
3. Find the equation of the line that goes through the following points,
leaving your equation in the form π¦ = ππ₯ + π.
a. (2,3), (6,7)
b. (β1,3), (4, β7)
c. (4,5), (β2,2)
d. (3,7), (9, 5)
4. Determine the equation of this line.
5. A line passes through the points (2,5) and (9, 10).
a) Find the equation of the line in the form ππ₯ + ππ¦ + π = 0, where
π, π, π are integers.
b) Hence determine the coordinate of the point where the line crosses
the π₯-axis.
6. The line π1 passes through the points π΄(15,11) and π΅(21,9) and
intercepts the π¦-axis at the point πΆ. The line π2 passes through πΆ and
π·(5,17). Determine the equation of the line π2 in the form π¦ = ππ₯ + π.
7. A line passes through (4, π + 13), (π, 4π + 1) for some constant π.
Determine the gradient of the line.
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Exercise 4: Distances between points and points of intersection
1. Find the coordinate of the point of intersection between these lines:
a. π¦ = π₯ + 5, π¦ = 2π₯
b. π¦ = 2π₯ β 5, π¦ = π₯ + 5
c. π₯ + π¦ = 5, π¦ = 2π₯ β 4
d. 2π₯ + π¦ = 7, π₯ β 2π¦ = 6
e. 4π₯ + 3π¦ = 1, π¦ = 1 β π₯
2. Find the distance: (giving exact values)
a) π΄π΅ b) π΄πΆ
c) πΆπ· e) π·πΈ
f) πΆπΈ
3. Find the distance between the two points where π¦ = 3π₯ + 12 crosses the
coordinate axes.
4. Line π1 passes through (β1,1) and (6,15). Another line π2 passes through
(0, β12) and (3,3). Determine the coordinate of the point at which they
intersect.
5. Line π1 has the equation π¦ = π₯ and π2 has the equation π¦ = β2π₯ + 12.
The two lines intersect at point π΄ and line π2 intersects the π₯ and π¦-axis at
π΅ and πΆ respectively, as indicated. Find the area of:
a. ππ΄π΅ (where π is the origin)
b. ππ΄πΆ
6. [AQA IGCSEFM Jan 2013 Paper 1 Q16] π΄, π΅ and πΆ are points on the line
2π₯ + π¦ = 8. π·πΆπΈ is a straight line.
π΄π΅: π΅πΆ = 2: 1
πΈπΆ: πΆπ· = 1: 2
Work out the ratio:
π΄πππ ππ π‘πππππππ π΄πΈπΆ:π΄πππ ππ π‘πππππππ π΅πΆπ·
Give your answer in its simplest form.
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Exercise 5: Parallel and perpendicular lines
1. Are the following lines parallel, perpendicular or neither?
a. π¦ = 2π₯ + 3, π¦ = 2π₯
b. π¦ = 3π₯ β 4, π¦ = β3π₯ + 1
c. π¦ =1
2π₯ + 1, π¦ = β2π₯
2. A line is parallel to π¦ = 2π₯ + 3 and goes through the point (4,3). What is
its equation?
3. A line π1 goes through the indicated point and is perpendicular to another
line π2. Determine the equation of π1 in each case.
a. (2,5) π2: π¦ = 2π₯ + 1
b. (β6,3) π2: π¦ = 3π₯
c. (0,6) π2: π¦ = β1
2π₯ β 1
d. (β9,0) π2: π¦ = β1
3π₯ + 1
e. (10,10) π2: π¦ = β5π₯ + 5
4. π΄(2,5) π΅(4,9)
Find the equation of the line which passes through B, and is perpendicular
to the line passing through both A and B.
5. Line π1 has the equation 2π¦ + 3π₯ = 4. Line π2 goes through the points
(2,5) and (5,7). Are the lines parallel, perpendicular, or neither?
6. Determine the equation of the line π.
7. Determine the equation of the line π.
8. π΄(3,7), π΅(5,13)
Find the equation of the line passing through π΅ and is perpendicular to
the line passing through π΄ and π΅, giving your answer in the form
ππ₯ + ππ¦ + π, where π, π, π are integers.
9. [AQA IGCSEFM June 2012 Paper 1 Q11] ππ΄π΅πΆ is a kite.
a. Work out the equation of π΄πΆ.
b. Work out the coordinates of π΅.
10. Suppose π is the origin, and π΄(1,2), π΅(4,2), πΆ(2.2, β 0.4).
Prove that ππ΄π΅πΆ is a kite.
(Hint: you need to prove two things as part of this.)
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Exercise 6: Mixed Exercises
1. Line π1 passes through the points (4,5) and (7,11). Line π2 has the
equation 2π¦ = 3π₯ β 1. Do the lines intersect?
2. π΄ is the point (4,β1) and π΅ is the point (7,7).
a. Find the coordinates of the midpoint of π΄π΅.
b. Find the distance π΄π΅ to 2 dp.
3. Line π1 has the equation π¦ = 2π₯ + 1 and line π2 the equation π¦ = 4π₯ β 3.
Find the coordinates of the point at which they intersect.
4. a) Find the gradient of the line with equation 3π₯ β 4π¦ = 12.
b) Prove that 3π₯ β 4π¦ = 12 and 3π¦ = 12 β 4π₯ are perpendicular.
5. A line passes through the points (0,4) and (6,1). Find the equation of the
line in the form:
a. π¦ = ππ₯ + π b. ππ₯ + ππ¦ = π where π, π, π are integers.
6. Find the coordinates of the points where 2π₯ β 3π¦ = 6 crosses:
a. The π₯-axis.
b. The π¦-axis.
7. [Edexcel] π΄π΅πΆπ· is a square. π and π· are points on the π¦-axis. π΄ is a point
on the π₯-axis. ππ΄π΅ is a straight line. The equation of the line that passes
through the points π΄ and π· is π¦ = β2π₯ + 6. Find the length of ππ·.
8. Determine the equation of this line, putting your answer in the form ππ₯ +
ππ¦ + π = 0, where π, π, π are integers.
9. A triangle consists of the points π(3, π), π(6,8) and π (10,10). Angle
πππ is a right angle.
Determine the equation of the line passing through π and π , leaving your
answer in the form ππ₯ + ππ¦ = π, where π, π, π are integers.