Video Transcript
Using the graph, determine
which of the following is a sensible estimate for the solution to the
simultaneous equations two 𝑥 plus three 𝑦 equals 20, four 𝑥 minus four 𝑦
equals 11. (a) 𝑥 equals 5.4 and 𝑦 equals
3.1. (b) 𝑥 equals 5.4 and 𝑦 equals
2.9. (c) 𝑥 equals 5.6 and 𝑦 equals
2.4. (d) 𝑥 equals 5.7 and 𝑦 equals
2.9. Or (e) 𝑥 equals 5.9 and 𝑦
equals 2.7.
Now, It may not be immediately
obvious, but the two lines we’ve been given on the graph do represent the two
equations given in the question. The red line is two 𝑥 plus
three 𝑦 equals 20, and the blue line is four 𝑥 minus four 𝑦 equals 11. The solution to this pair of
simultaneous equations then will be the coordinates of the point where these two
lines intersect. But from looking at the figure,
we can see that they intersect in the middle of one of the smaller squares. So, we can’t find an exact
value for the solution. Instead, we’re going to be
looking for an estimate.
Let’s first make sure we’re
clear on the scale that has been used on each of our axes. In each case, it’s the
same. There are four small squares to
represent two units. Dividing by four, we see that
each small square on each axes represents 0.5 units. Looking at the horizontal
placement of this point first of all then, we can see that it is located between
the line three small squares to the right of four and then the 𝑥-value six. If each small square is 0.5,
then three small squares is 1.5, which means that the 𝑥-value to the left of
this point is 5.5. And so, our 𝑥-value is between
5.5 and six.
In the same way, looking at the
vertical placement of this point, we can see that it’s located between one small
square above two, that’s 2.5, and two small squares above two, that’s three. So, the 𝑦-value is between 2.5
and three.
Looking at the five options, we
can rule out options (a) and (b), as their 𝑥-values are out of range, and
option (c), as its 𝑦-value is out of range. To decide between the two
remaining options then, we know that the point is vertically very close to
three, and it seems to be closer to 5.5 than to six. So, option (d) is the most
sensible estimate. 𝑥 is approximately equal to
5.7, and 𝑦 is approximately equal to 2.9.
