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Standard Level

3.05 Find and use the equation of a line

Worksheet
Gradient-intercept form
1

Consider the equation of a line y = mx + c.

a
What does m represent?
b
What does c represent?
2

Consider the following three linear equations with their graphs plotted on a number plane:

  • Equation 1: y = x + 1

  • Equation 2: y = 2 x + 1

  • Equation 3: y = 4 x + 1

a

What do all of the equations have in common?

b

What do all of the graphs have in common?

c

Describe all lines that have the form: y = m x + 1

-2
-1
1
2
x
-2
-1
1
2
y
3

Consider the following three linear equations with their graphs plotted on a number plane:

  • Equation 1: y = 2 x + 4

  • Equation 2: y = 2 x + 8

  • Equation 3: y = 2 x - 4

a

What do all of the equations have in common?

b

What do all of the graphs have in common.

c

Describe all lines that have the form: y = 2 x + c

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
4

Determine whether or not the gradients of the following pairs of equations are equal:

a
i
y=7x
ii
y=7x-4
b
i
y=7x+2
ii
y=\dfrac{x}{7} +4
c
i
y=6-7x
ii
y=6x+7
d
i
y=4+7x
ii
y=7x-10
5

Consider the line plotted below:

a

State the the y-intercept.

b

State the gradient.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
6

Consider the line plotted below:

a

By how much does the y-value change as the x-value increases by 1?

b

State the gradient of the line.

-1
1
2
3
x
-2
-1
1
2
3
4
5
6
7
8
9
10
y
7

State whether or not the following lines have a y-intercept:

a

y = 2

b

5 x = 4 y

c

y = - 4 x + 1

d

x = 1

e
y = 6 x - 2
f

y = 3 x

g
2 x + 6 y = 3
h
y + 4 = -5 x
8

Consider the following linear equations:

i

State the value of the gradient, m.

ii

State the y-intecept, c.

a

y = - 2 x + 9

b

y = - 4 x - 8

c

y = 8 x + 6

d

y = - 1 - \dfrac{9 x}{2}

e
3y = 12x -15
f

- 9 x + 9 y = 27

g

3 x - 10 y =- 2

h
16x + 12y = 10
9

Sketch the following lines on a number plane:

a

The line with a y-intercept of - 2 and gradient of - 3.

b

The line with a y-intercept of 3 and gradient of - \dfrac{3}{2}.

c

The line y = 2 x + 5.

d

The line y = \dfrac{1}{2} x - 1.

10

For each of the following equations:

i

Find the y-value of the y-intercept of the line.

ii

Find the x-value of the x-intercept of the line.

iii

Find the value of y when x = 3.

iv

Sketch the line on a number plane.

a
y = - 2 x
b
y = \dfrac{2 x}{3} - 4
11

For each of the following lines:

i

Find the y-coordinate of the y-intercept of the line.

ii

Hence, write the equation of the line in gradient-intercept form.

iii

Find the x-coordinate of the x-intercept of the line.

iv

Sketch the line on a number plane.

a

A line has gradient \dfrac{4}{5} and passes through the point \left( - 10 , 4\right).

b

A line has gradient - 2 and passes through the point \left(3, - 8 \right).

Equation of a line
12

A line has a gradient of - 3 and intercepts the y-axis at - 2.

a

Find the equation of the line in the form y = m x + c.

b

State whether the point \left( - 2 , 4 \right) lies on this line.

13

A line has a gradient of - 3 and cuts the y-axis at 8.

a

Find the equation of the line in the form y = m x + c.

b

State whether the point \left( 8, - 31 \right) lies on this line.

14

Find the equations of the following in the form y = mx + c :

a

A line that has the same gradient as y = 9 - 8 x and the same y-intercept as

y = - 5 x - 3.

b

A line whose gradient is 2 and crosses the y-axis at 5.

c

A line whose gradient is - 6 and crosses the y-axis at 9.

d

A line whose gradient is - 8 and crosses the y-axis at - 9.

e

A line whose gradient is - \dfrac{3}{4} and intercepts the y-axis at 3.

f

A line whose gradient is \dfrac{4}{3} and goes through the point \left(0, 3\right).

g

A line that has a gradient of - 2 and passes through \left( - 6 , - 3 \right)

h

A line whose gradient is 8 and goes through the point \left(0, - 4 \right).

i

A line whose gradient is 0 and goes through the point \left(0, \dfrac{2}{13}\right).

15

Consider the line plotted on the number plane.

a

State the values of:

i

The gradient, m.

ii

The y-intercept, c.

b

Find the equation of the line in gradient-intercept form.

c

Find the value of y when x = 27.

-2
-1
1
2
3
x
-2
-1
1
2
3
4
5
y
16

Consider the line plotted on the number plane.

a

Find the equation of the line in gradient-intercept form.

b

Find the value of y when x = 50.

-1
1
2
3
x
-5
-4
-3
-2
-1
1
y
17

Consider the line plotted on the number plane.

a

Find the equation of the line in gradient-intercept form.

b

Find the value of y when x = 29.

-2
-1
1
2
x
-2
-1
1
2
y
18

Find the equations of the following lines in gradient-intercept form:

a
-1
1
2
3
x
-5
-4
-3
-2
-1
1
y
b
-2
2
4
6
8
10
12
14
16
18
x
-4
-3
-2
-1
1
y
c
-1
1
2
3
4
5
x
-3
-2
-1
1
y
d
-6
-5
-4
-3
-2
-1
1
x
-1
1
2
3
y
e
-2
-1
1
2
x
-2
-1
1
2
y
f
-1
1
2
3
x
-2
-1
1
2
3
4
5
6
y
g
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
h
-2
-1
1
2
x
-4
-3
-2
-1
1
2
y
19

For the following lines passing through the given two points:

i

Hence, state the gradient of the line.

ii

Find the equation of the line in the form y = m x + c.

a

\left(0, 2\right) and \left(2, 6\right)

b

\left(0, - 9 \right) and \left(5, 1\right)

c

\left(0, 2\right) and \left( - 7 , 44\right)

d
\left(0, - 3 \right) and \left(1, 4\right)
20

For each of the following linear equations:

i

Rewrite it in the form y = m x + c.

ii

State the gradient of the line, m.

iii

State the y-intercept of the line, c.

a

y = \dfrac{- 4 x + 16}{4}

b

9 x - y - 8 = 0.

c

y = 3 \left( 4 x - 3\right)

d

y = 6 \left( 3 x - 2\right)

e

3 x - 9 y - 27 = 0.

f

3 x - 4 y - 28 = 0

21

Consider the lines with the following equations:

  • Line A: 5 x + 3 y + 5 = 0

  • Line B: 7 x + 6 y - 3 = 0

a

Express the lines in the form y = m x + c.

b

State which line is steeper, A or B.

22

Determine which of the following lines are steeper: 2 x + 5 y - 5 = 0 or 4 x + 4 y + 1 = 0.

23

A straight line has gradient -1 and goes through the points \left(0, 2\right) and \left(a, - 6 \right).

a

Write the equation of the line in the form y = m x + b.

b

Find the value of a.

Gradients of parallel lines
24

Determine whether the following statements about two parallel lines are true or false.

a

The y-value is changing at the same rate on both lines.

b

They intersect at one point.

c

They have the same value of c in y = m x + c.

d

They have the same value of m in y = m x + c.

e

They are equidistant from each other.

25

State whether the given pairs of lines are parallel:

a

y = - 2 x - 5 and y = - 2 x - 8

b

y = 7 x + 8 and y = - 5 x + 8

c

y = - 3 x - 2 and y = - 3 x + 9

d

y = - 6 x - 5 and y = - 6 x

26

Find the gradient of the following lines:

a

A line parallel to a line with gradient - 2.

b

Any line that is parallel to y = - 7 + 4 x.

27

State whether the following lines are parallel to y = 7 x + 3.

a

y = 7 x - 3

b

y = 6 x + 3

c

y = 7 x

d

y = - 7 x + 3

28

State whether the following lines are parallel to y = - 3 x + 2.

a

y = 3 x

b

y = - \dfrac{2 x}{3} + 8

c

- 3 y - x = 5

d

y = - 10 - 3 x

Point-gradient formula
29

For the following graphs:

i

State the value of the x-intercept.

ii

State the value of the y-intercept.

a
-1
1
2
3
4
x
-1
1
2
3
4
y
b
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
c
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
y
d
-4
-3
-2
-1
1
2
3
4
5
6
x
-1
1
2
3
4
y
e
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
f
-2
-1
1
2
3
4
x
-5
-4
-3
-2
-1
1
2
y
30

For each of the following tables of values:

i

Find the gradient, m.

ii

Find the y-intercept, c.

iii

Write the equation of the line expressing the relationship between x and y.

iv

Complete the table of values.

a
x0123424
y0481216
b
x0123421
y914192429
c
x0123425
y- 23- 21- 19- 17- 15
d
x0123470
y272217127
31

For each of the following tables of values:

i

Find the equation of the line expressing the relationship between x and y.

ii

Complete the table of values.

a
x123419
y5101520
b
x123416
y- 3- 6- 9- 6
c
x123460
y14710
d
x123480
y- 14- 22- 30- 38
e
x1234-15
y82746658
32

Find the equation that corresponds to each of the following tables:

a
x- 10123
y741- 2- 5
b
x12345
y58111417
c
x- 8- 7- 6- 5- 4
y- 37- 32- 27- 22- 17
d
x34567
y-1- 3- 5- 7- 9
33

Find the equation of the following lines:

a

A line that passes through the point A \left( - 5 , - 4 \right) and has a gradient of - 4.

b

A line that passes through the point A \left( - \dfrac{4}{5} , - 4 \right) and has a gradient of 2.

c

A line that passes through Point A \left( - 4 , 3\right) and has a gradient of 4.

d

A line that passes through Point A \left(7, - 6 \right) and has a gradient of - 3.

e

A line that passes through the point A \left(3, 5\right) and has a gradient of - \dfrac{5}{2}.

f

A line that passes through the point A \left(4, 3\right) and has a gradient of - 3\dfrac{1}{3}.

g

A line that passes through the point A \left( - 4 , 3\right) and has a gradient of - 9.

h

A line that passes through the point A \left( - \dfrac{5}{9} , 7\right) and has a gradient of 7.

i

A line that passes through the point A \left(8, 1\right), and has a gradient of \dfrac{5}{2}.

j

A line that passes through the point A \left( - 4 , 5\right) and has a gradient of 3\dfrac{1}{2}.

34

For each of the following lines:

i

Find the equation of the line.

ii

Sketch the graph of the line.

a

A line has gradient 2 and passes through the point \left( - 5 , - 3 \right).

b

A line has gradient - \dfrac{3}{2} and passes through the point (- 2, 2).

c

A line has gradient - \dfrac{2}{5} and passes through the point \left( - 10 , 2\right).

d

A line has gradient - 3 and passes through the point \left(2, - 12 \right).

35

Consider the line with equation 2 x + y - 8 = 0.

a

Find the x-intercept of the line.

b

Hence, find the equation of a line with a gradient of - 4 that passes through the x-intercept of the given line.

36

Find, in general form, the equation of a line which has a gradient of \dfrac{4}{7} and cuts the x-axis at - 10.

37

For each of the following lines:

i

Find the gradient of the line.

ii

Find the equation of the line.

a

A line passes through the points \left(2, - 7 \right) and \left( - 5 , 6\right).

b

A line passes through the points \left(3, - 3 \right) and \left(5, - 11 \right).

c

A line passes through the points A \left( - 6 , 7\right) and B \left( - 8 , - 4 \right).

38

Identify which of the following equations of straight lines have a gradient of 5 and pass through the point A \left(-1, - 4 \right):

a

\dfrac{y + 4}{x + 1} = 5

b

\dfrac{x + 1}{y + 4} = 5

c

\dfrac{- 4 - y}{-1 - x} = 5

d

\dfrac{y + 1}{x + 4} = 5

39

Write down the equations of three lines that pass through the point (1, 3). Explain how your lines are different.

Applications
40

A line has a gradient of \dfrac{3}{10} and passes through the midpoint of A \left( - 6 , - 6 \right) and B \left(8, 8\right).

a

Find the coordinates of M, the midpoint of AB.

b

Find the equation of the line in general form.

41

Consider the lines L_{1}, y = - 4 x + 5, and L_{2}, y = x - 1.

a

Find the midpoint M of their y-intercepts.

b

Find the equation of the line that goes through the point M and has gradient \dfrac{1}{3}. Express the equation in general form.

42

A circle with centre C \left(11, 13\right) has a diameter with end points A \left(5, 14\right) and B \left(p, q\right).

a

Find the value of p.

b

Find the value of q.

c

Find the equation of the line passing through B with gradient \dfrac{9}{2}.

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