Further straight line problems
You know a number of ways to draw lines. One quick method is to find
its x- and y-intercepts from its equation.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Draw a quick sketch of the line 3x – 2y + 6 = 0
Solution
On the x-axis, y = 0.
On the y-axis, x = 0.
3x 2 y + 6 = 0
3x 2(0) + 6 = 0
3x 2 y + 6 = 0
3(0) 2 y + 6 = 0
3x + 6 = 0
3x = 6
x = 2
2 y + 6 = 0
2y = 6
y=3
You now have enough information to draw your line.
Part 1
Equations of a straight line
1
While this is a quick method of drawing a line,
it only relies on two points. So you need to be
careful that your points are correct.
With a bit of practice you might be able to
determine the x- and y-intercepts mentally.
To find the equation of a straight line you must know two things about it.
If you know
•
one point (x1 , y1 ) and the gradient m, the equation is
y y1 = m(x x1 )
•
two points (x1 , y1 ) and (x2 , y2 ) , the equation is
y y1
x x1
=
y2 y1
.
x2 x1
Sometimes the information you are given may not be as obvious as this.
Follow through the steps in this example. Do your own working in the
margin if you wish.
Find the equation of the line which passes through the point
(2, –3) and is parallel to the line with equation 2x + y = 3.
Solution
As the line is parallel to 2x + y = 3, both lines have the same
gradient.
2x + y = 3
y = 2x + 3
The gradient of the line is m = –2.
y y1 = m(x x1 )
where m = 2, (x1 , y1 ) = (2, 3)
y (3) = 2(x 2)
y + 3 = 2x + 4
y = 2x +1
2x + y 1 = 0
2
(gradient/intercept form)
(general form)
PAS5.3.3 Coordinate geometry
Activity – Further straight line problems
Try these.
1
a
At what points does the line 4x + 3y + 8 = 0 intercept the axes?
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b
Draw the graph of this line on the grid.
5
y
4
3
2
1
–5
–4
–3
–2
–1
0
1
2
3
4
5x
–1
–2
–3
–4
–5
2
A line is parallel to the line 2x + 3y + 9 = 0 and passes through (2, 1).
a
What is the equation of this line?
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Part 1
Equations of a straight line
3
b
The line on the grid below shows 2x + 3y + 9 = 0. Include the
graph of the line parallel to it and passing through (2, 1).
4
y
3
2
1
–5
–3
–4
–2
–1
0
1
2
3
x
4
–1
–2
–3
–4
–5
3
The graph below shows the lines bounding three sides of a
parallelogram.
4
y
3
2
1
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5 x
–1
–2
–3
–4
–5
Write the equation of the fourth line, passing through (–2, –4) which
completes the enclosed parallelogram.
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4
PAS5.3.3 Coordinate geometry
Check your response by going to the suggested answers section.
Part 1
Equations of a straight line
5
Activity – Further straight line problems
1
a
On the x-axis, y = 0
On the y-axis, x = 0
4x + 3y + 8 = 0
4(0) + 3y + 8 = 0
4x + 3y + 8 = 0
4x + 3(0) + 8 = 0
3y = 8
2
8
y = (or y = 2 )
3
3
4x = 8
x = 2
3
b
y
2
1
–4
–3
–2
–1 0
1
2 x
–1
–2
–3
–4
–5
2
a
2x + 3y + 9 = 0
3y = 2x 9
2
y = x3
3
2
As m1 = m2 , then m = .
3
y y1 = m(x x1 )
2
y 1 = (x 2)
3
3y 3 = 2x + 4
2x + 3y 7 = 0
5
b
y
4
3
2
1
–5
–4
–3
–2
–1
0
–1
1
2
3
4
5x
–2
–3
–4
–5
6
PAS5.3.3 Coordinate geometry
3
The line opposite the missing side has gradient m =
3 1
= .
6 2
y y1 = m(x x1 )
1
y + 4 = (x + 2)
2
2y +8 = x + 2
x 2y 6 = 0
Part 1
Equations of a straight line
7