Video Transcript
Given that the ordered pair
negative seven, negative three satisfies the relation three 𝑥 plus 𝑏𝑦 is equal to
negative three, find the value of 𝑏.
To find the value of 𝑏 in the
linear relation given, we substitute the 𝑥- and 𝑦-values from the ordered pair
into the linear relation. We then solve the linear relation
for 𝑏. With 𝑥 is equal to negative seven
and 𝑦 is negative three then, into the linear relation three 𝑥 plus 𝑏𝑦 is
negative three, this gives us three times negative seven plus 𝑏 times negative
three is equal to negative three. That is, negative 21 minus three 𝑏
is equal to negative three. To solve for 𝑏, we can then add 21
to both sides, and this gives negative three 𝑏 is equal to negative three plus 21,
which evaluates on the right-hand side to 18.
And now dividing both sides by
negative three, we have 𝑏 is equal to 18 over negative three. That is, 𝑏 is equal to negative
six. Hence, the value of 𝑏 that
satisfies the relation three 𝑥 plus 𝑏𝑦 is equal to negative three given that the
point negative seven, negative three satisfies this relation is 𝑏 is negative
six.
A linear relation can be
represented graphically as a straight line, hence the term “linear.” If we know at least two ordered
pairs that satisfy a specific linear relation, to represent the relation
graphically, we plot the points represented by the ordered pairs and draw the line
that passes through both points. For example, the ordered pairs
negative one, negative three and two, three both satisfy the linear relation
negative two 𝑥 plus 𝑦 is equal to negative one. We can represent this relation
graphically by plotting the points with coordinates 𝑥 is equal to negative one, 𝑦
is negative three and 𝑥 is two, 𝑦 is three and drawing a line through those
points.
Note that while we’ve plotted the
line corresponding to the linear relation using only the two points given, in fact,
every point on the line is represented by an ordered pair 𝑥, 𝑦 that satisfies the
linear relation negative two 𝑥 plus 𝑦 is equal to negative one. And that’s where 𝑥 and 𝑦 are the
coordinates of the point on the line.
With this in mind, looking once
again at our postage stamps example, we note that the linear relation 20𝑥 plus 50𝑦
is equal to 240, which we can write equivalently as two 𝑥 plus five 𝑦 is 24, can
be represented by the graph shown. In this case, however, while the
linear relation 20𝑥 plus 50𝑦 is equal to 240 is represented completely by the
plotted line, we know that, in fact, there are only three points on that line that
correspond to the actual scenario of buying stamps to the value of 240 piaster. Remember that 𝑥 was the number of
20-piaster stamps and 𝑦 the number of 50-piaster stamps and these cannot be broken
down into smaller units. And so for this particular
scenario, the solutions must be positive whole-number values. While this is true for the number
of stamps, however, any point 𝑥, 𝑦 on the given line satisfies the linear relation
20𝑥 plus 50𝑦 is 240, or equivalently two 𝑥 plus five 𝑦 is equal to 24.
Now, in the linear relations
discussed so far, that is, relations of the form 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑐, the
coefficients 𝑎 and 𝑏 have been nonzero. Let’s look now at the special cases
where either 𝑎 or 𝑏 is equal to zero.
In the linear relation 𝑎𝑥 plus
𝑏𝑦 is equal to 𝑐, in the special case where 𝑎 is equal to zero, the relation
reduces to 𝑏𝑦 is equal to 𝑐. If we then divide through by 𝑏, we
have 𝑦 is equal to 𝑐 over 𝑏. This means that for every value of
𝑥, 𝑦 is equal to the constant 𝑐 over 𝑏. What this means graphically is that
this is a horizontal line through the point on the 𝑦-axis where 𝑦 is equal to 𝑐
over 𝑏. So this is the case when 𝑎 is
equal to zero.
