Two intersecting straight lines are
shown. Find the values of 𝑥 and 𝑦.
As the question says, we have a
pair of intersecting straight lines. And when this is the case, we need
to look out for vertically opposite angles. Well, we have two pairs of
those. These two angles are vertically
opposite from one another. And the angle labeled as 𝑦 is
vertically opposite to this one. In fact, though, we’re just going
to begin by worrying about 𝑥.
Next, we state the following
fact. Vertically opposite angles are
equal. Now, we said that the angle nine 𝑥
minus 30 degrees is vertically opposite to the angle seven 𝑥 minus four
degrees. This must mean then that we can say
that nine 𝑥 minus 30 is equal to seven 𝑥 minus four. Our job now is to solve this
equation for 𝑥.
Now, since we have 𝑥s on both
sides of our equation, our first job is to get rid of the smallest number of 𝑥. Seven 𝑥 is less than nine 𝑥, so
we’re going to subtract seven 𝑥 from both sides of our equation. Nine 𝑥 minus seven 𝑥 is two
𝑥. So, the left-hand side of our
equation is two 𝑥 minus 30. On the right-hand side, we’re
simply left with negative four.
Next, we want to get rid of the
negative 30. So, we do the inverse operation; we
add 30 to both sides. On the left-hand side, that leaves
us with two 𝑥. And negative four plus 30 is
positive 26. So, our equation is two 𝑥 equals
26. Remembering, of course, that two 𝑥
means two times 𝑥, we know now that we need to divide both sides of the equation by
two. Two 𝑥 divided by two is 𝑥. And 26 divided by two is 13.
So, we’ve calculated 𝑥 to be equal
to 13. We’re not quite finished
though. The question also asks us to find
the value of 𝑦. So, what do we do next? Well, there are a couple of things
we could do. One of the facts we could use is
that angles on a straight line sum to 180 degrees. We could also use the fact that
angles about a point sum to 360 degrees. But, here, that requires a little
more work. So, we’re going to calculate the
value of either seven 𝑥 minus four or nine 𝑥 minus 30.
Remember, that will give us the
same value. Let’s calculate seven 𝑥 minus four
when 𝑥 is equal to 13. It’s seven times 13 minus four,
which is 87. So, this angle here is 87
degrees. We can set up and solve an equation
for 𝑦. We know that angles on a straight
line sum to 180 degrees, so we say that 87 plus 𝑦 equals 180. This time, we solve by subtracting
87 from both sides. And we see that 𝑦 is equal to
93. 𝑥 is equal to 13, and 𝑦 is equal
Now, in fact, with angle problems,
there’s often more than one way to solve that same problem. In this case, we could have
originally started by quoting the fact that angles on a straight line sum to 180
degrees. We form one equation by adding
these two angles. And we know that equals 180. So, we get nine 𝑥 minus 30 plus 𝑦
equals 180. Now, let’s combine the numerical
parts by adding 30 to both sides. And when we do, we find that nine
𝑥 plus 𝑦 equals 210.
Next, we use the same idea, this
time adding these two angles. We get seven 𝑥 minus four plus 𝑦
equals 180. To combine the numerical parts, we
add four to both sides. And we find that seven 𝑥 plus 𝑦
equals 184. Notice, we now have a pair of
simultaneous equations. And we can see that the
coefficient, the number of 𝑦s we have in each equation, is the same. Since the signs of the coefficient
of 𝑦 is also the same, we subtract both equations.
Nine 𝑥 minus seven 𝑥 is two
𝑥. 𝑦 minus 𝑦 is zero. And 210 minus 184 is 26. To solve for 𝑥, we divide through
by two. And we once again find 𝑥 to be
equal to 13 degrees. We find the value of 𝑦 by
substituting 𝑥 into either of our original equations. If we substitute it into the second
one, we get seven times 13 plus 𝑦 equals 184. Seven times 13 is 91. And then, we solve for 𝑦 by
subtracting 91 from both sides. And once again, we find 𝑦 to be
equal to 93.