2) The straight-line graph
y  3x  1
intersects (i.e. crosses) the straight-line graph
y  9  2x
when x = 2 and y = 5, i.e. at coordinates (2,5).
To help understand the answer to this question we return first to some basic points about
graphs.
y-axis
To plot a graph we first need to draw axes.
origin 0
x-axis
The horizontal axis is the x-axis and the vertical axis is the y-axis. The intersection of these axes is the
origin. Axes should always be labelled.
The number of units represented by a unit length along an axis is called the scale.
e.g. 1 cm = 2 units
We can have different scales on each axis.
Coordinates mark the points on the graph. For example the point plotted as shown has the
value x = 4 and y = 6 and is said to have coordinates (4,6). The x value is always given first.
8
6
y4
2
0
(4,6)
0
1
2
3
4
5
x
The equation of a straight line can be written in the following form:
y  mx  c
where m is the gradient (i.e. how steep the line is) and c is the intercept (i.e. the point at which the
line crosses the y axis).
The graph of y = 3x - 1 has gradient 3 (m = 3) and intercept -1 (c = -1).
The graph of y = 9 - 2x has gradient -2 (m = -2) and intercept 1 (c = 9).
Draw these two graph lines on the same set of axes to see where they cross.
To draw a graph of y = 3x - 1 for values of x between –3 and 3, consider completing the following
table to help you work through the different steps.
x
-3
-2
-1
0
1
2
3
3x
-9
-6
-3
0
3
6
9
-1
-1
-1
-1
-1
-1
-1
-1
y
-10
-7
-4
-1
2
5
8
From this table you can read off coordinates (x, y) from the top and bottom rows, to plot on your
paper, and then join up the dots or crosses with a straight line:
(-3,-10)
(-2,-7)
(-1,-4)
(0,-1)
(1,2)
(2,5)
(3,8)
To draw a graph of y = 9 - 2x for values of x between –3 and 3, consider completing the following
table to help you work through the different steps.
x
-3
-2
-1
0
1
2
3
-2x
6
4
2
0
-2
-4
-6
+9
+9
+9
+9
+9
+9
+9
+9
y
15
13
11
9
7
5
3
From this table you can read off coordinates (x, y) from the top and bottom rows, to plot on your
paper, and then join up the dots or crosses with a straight line:
(-3,15)
(-2,13)
(-1,11)
(0,9)
(1,7)
(2,5)
(3,3)
Note: the line y = 3x - 1 slopes upwards since it has a positive gradient of 3 whereas the line y = 9 - 2x
slopes downwards since it has a negative gradient of -2.
The point at which these two graph lines cross is when x = 2 and y = 5, i.e. at coordinates (2,5). You
should be able to see this clearly from plotting these two graph lines on the same set of axes.
However, we are lucky in this example since the point (2,5) appears in both coordinate lists above and
so we can already infer from this information that the two graph lines must intersect at that point, even
without drawing the graphs!
A quicker alternative method for identifying this point of intersection is to think through what it
means mathematically to have two graph lines crossing.
At the point of intersection, the value of x must be the same for both lines, and the value of y must
also be the same for both lines.
This means we can set the right-hand-side of each line equation to equate.
When
y  3x  1
and
y  9  2x
cross, the y values must be the same:
3x  1  9  2x
We can then solve this equation for x by collecting terms in x on the left-hand-side:
3x  2x  9  1
5x  10
10
5
x2
x
Now that we know the value of x at the crossing point, we can substitute this into either line equation
in order to identify the value of y at the crossing point.
y  3x  1
y  3  2  1
y  6 1
y5
We can also check we get the same value of y when we substitute x = 2 into the other line
equation:
y  9  2x
y  9  2  2
y 94
y5
In this way we have shown that these two graph lines cross when x = 2 and y = 5, i.e. at
coordinates (2,5).