A line 𝐿 one has equation 𝑦
equals two 𝑥 minus five and a line 𝐿 two has equation two 𝑦 plus four 𝑥 equals
minus seven or negative seven. Determine whether the two lines are
Two lines are perpendicular if 𝑚
one multiplied by 𝑚 two is equal to minus one if their gradients multiply together
to give us an answer of minus one. 𝐿 one, 𝑦 equals two 𝑥 minus
five, is already written in the form 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑚 is the
gradient and 𝑐 is the 𝑦-intercept. This means that the gradient of 𝐿
one is equal to two. 𝑚 one is equal to two.
In order to work out the gradient
of 𝐿 two, we need to make 𝑦 the subject of the equation so that it is in the form
𝑦 equals 𝑚𝑥 plus 𝑐. Subtracting four 𝑥 from both sides
of the equation gives us two 𝑦 is equal to minus four 𝑥 minus seven. Dividing both sides of this
equation by two gives us 𝑦 is equal to minus two 𝑥 minus seven over two or minus
3.5. The gradient of 𝐿 two is,
therefore, minus two.
We know that two lines are
perpendicular if 𝑚 one multiplied by 𝑚 two is equal to minus one. In this case, 𝑚 one was equal to
two and 𝑚 two was equal to minus two. Two multiplied by minus two is
equal to minus four. As the two gradients do not have a
product of minus one, they are not perpendicular.
The lines 𝐿 one, 𝑦 equals two 𝑥
minus five, and 𝐿 two, two 𝑦 plus four 𝑥 equals minus seven, are not