Video Transcript
By plotting the graphs of 𝑦
equals negative two 𝑥 plus one and 𝑦 equals 𝑥 plus four, find the point that
satisfies both equations simultaneously.
We’re told in this problem that
we need to approach it by plotting the graphs of these two equations. So, let’s consider two
different methods for plotting straight-line graphs. In our first method, we’ll plot
the graph of 𝑦 equals negative two 𝑥 plus one by comparing its equation to the
general equation of a straight line in its slope–intercept form, 𝑦 equals 𝑚𝑥
plus 𝑏. We recall that, in this form,
the value of 𝑚, that’s the coefficient of 𝑥, represents the slope of the
line. So, the slope of the line we’re
looking to plot is negative two. This means that for every one
unit the line moves across to the right, it will move two units downwards.
We also recall that, in this
general form, the value of 𝑏, the constant term, represents the 𝑦-intercept of
the line. So in our equation, the
𝑦-intercept is positive one. This is the value at which the
line intersects the 𝑦-axis. Let’s plot this line then. We know that it intercepts the
𝑦-axis at a value of one. As the slope is negative two,
we know that if we move one unit across, we then need to move two units
downwards. So, we can plot our next point
at the coordinates one, negative one. We then move one across and two
down again and plot our next point at two, negative three.
We can also go back the other
way from our 𝑦-intercept. If we move one unit to the
left, we need to move two units up. So, we can also plot a point
with coordinates negative one, three. Joining all of these points
together with a straight line, and we have our first line 𝑦 equals negative two
𝑥 plus one.
We’ll use a different method to
plot the second line, which has equation 𝑦 equals 𝑥 plus four. This time, we’ll use a table of
values. We’ll choose a range of
different 𝑥-values. I’ve chosen the integer values
from negative two to two. And we’ll then use the equation
of the line to work out the corresponding 𝑦-values. For example, when 𝑥 is equal
to zero, 𝑦 will be equal to zero plus four, which is four. When 𝑥 is equal to negative
one, 𝑦 will be equal to negative one plus four or four minus one, which is
three.
In the same way, we can then
complete the rest of our table. We can then plot these points
or at least those which will fit on the axes I’ve got here and then join them
together with a straight line to give our second line 𝑦 equals 𝑥 plus
four. Notice that the 𝑦-intercept of
this line is four, and our line does indeed cross the 𝑦-axis at this point. And the slope is one, and our
line does indeed have a slope of one. So, we’ve plotted the two
lines. And now, we need to find the
point that satisfies both equations simultaneously. This is the point that lies on
both lines. It’s the coordinates of their
point of intersection.
From our graph, we can see that
the lines intersect at the point with coordinates negative one, three. As we we’re asked to find the
point that satisfies both equations simultaneously, we’ll give our answer as a
coordinate. So, our solution to the problem
is the coordinate negative one, three.
