Video Transcript
Which of the following sets of
simultaneous equations could be solved using the given graph? (a) 𝑦 equals two 𝑥 minus four
and 𝑦 equals 𝑥 plus five. (b) 𝑦 equals negative four 𝑥
plus two and 𝑦 equals five 𝑥 minus one. (c) 𝑦 equals two 𝑥 minus four
and 𝑦 equals negative 𝑥 plus five. (d) 𝑦 equals two 𝑥 plus four
and 𝑦 equals negative 𝑥 plus five. Or (e) 𝑦 equals negative four
𝑥 plus two and 𝑦 equals five 𝑥 plus one.
So, we’ve been given a graph of
two straight lines and asked to determine which pair of simultaneous equations
we could solve using this graph. This means that we need to
determine the equations of the two straight lines. In order to do this, we’ll
recall the general form of a straight line in its slope–intercept form 𝑦 equals
𝑚𝑥 plus 𝑏. And we recall that the
coefficient of 𝑥, that’s 𝑚, gives the slope of the line. And the constant term, that’s
𝑏, gives the 𝑦-intercept of the graph. That’s the 𝑦-value at which
the line intercepts the 𝑦-axis.
Let’s consider the blue line
first of all. We can see that this line
intercepts the 𝑦-axis at five, which means the equation of this line will be in
the form 𝑦 equals some number of 𝑥 plus five. To find the slope of this line,
the value of 𝑚, we can draw in a little right-angled triangle anywhere below
this line. And in doing so, we see that
for every one unit the line moves across, that’s to the right, it also moves one
unit down. As the slope of a line can be
found using change in 𝑦 over change in 𝑥, we have negative one over one, which
is negative one. The equation of this line is,
therefore, 𝑦 equals negative 𝑥 plus five.
So, we found the equation of
our first line. Let’s now consider the red
line. And this time we see that this
line intercepts the 𝑦-axis at a value of negative four. The equation of this line is,
therefore, in the form 𝑦 equals some number of 𝑥 minus four. To find the slope, we again
sketch in a right-angled triangle anywhere below this line. And this time, we see that for
every one unit the line moves to the right, it moves two units up. The slope of the line, change
in 𝑦 over change in 𝑥, is therefore two over one, which is equal to two. So, the equation of this line
is 𝑦 equals two 𝑥 minus four.
Looking at the five possible
options we were given, we can see that this combination of equations of straight
lines is option (c) 𝑦 equals two 𝑥 minus four and 𝑦 equals negative 𝑥 plus
five. Although we aren’t actually
asked to solve the pair of simultaneous equations in this question, we could do
so by looking at the coordinates of the point of intersection of the two
lines. And we see that the coordinates
of this point are three, two. So, the solution to this pair
of simultaneous equations would be 𝑥 equals three and 𝑦 equals two.
