Find the equation of the line that
passes through the points 𝐴: negative 10, two and 𝐵: zero, five, giving your
answer in the form of 𝑎𝑦 plus 𝑏𝑥 plus 𝑐 is equal to zero.
Now, we’ve been asked to find the
equation of this straight line in a specific form, with all of the terms on the same
side of the equation. But we’ll begin by using the
general form of the equation of a straight line. That’s 𝑦 equals 𝑚𝑥 plus 𝑏,
where 𝑚 is the slope of the line and 𝑏 is its 𝑦-intercept.
Now, it’s important to be aware
here that this value 𝑏 is not necessarily the same in our general equation and in
the requested form. We’re told that the line passes
through the point with coordinates zero, five. So the 𝑦-intercept of our line is
zero, five. And we can work out straightaway
that the value 𝑏 in our general form is five.
To work out the value of 𝑚, we can
use the formula change in 𝑦 over change in 𝑥. Subtracting the coordinates of 𝐴
from those of 𝐵, and we have five minus two over zero minus negative 10. We need to be really careful here
because we are subtracting a negative value. A common mistake would be to just
have zero minus 10 in the denominator. But that wouldn’t actually be
subtracting the 𝑥-coordinate of 𝐴 from the 𝑥-coordinate of 𝐵.
Simplifying, we have three in the
numerator. And in the denominator, zero minus
negative 10 is zero plus 10, which is 10. So the slope of the line is
three-tenths. Substituting the values of 𝑚 and
𝑏 into the equation of our straight line then, and we have 𝑦 equals three-tenths
of 𝑥 plus five.
Now, this isn’t in the required
form because we’re asked to collect all the terms on the same side of the
equation. And although the question doesn’t
explicitly say this, the values of 𝑎, 𝑏, and 𝑐 that we use should be
integers. We have a fraction of
three-tenths. But we can eliminate this by
multiplying each side by the denominator of 10. Doing so and remembering we need to
multiply every term in the equation, including the constant term, by 10, we get 10𝑦
equals three 𝑥 plus 50. Grouping all the terms on the
left-hand side by subtracting three 𝑥 and subtracting 50 gives the equation 10𝑦
minus three 𝑥 minus 50 equals zero, which is in the requested form.
Alternatively, we could’ve grouped
the terms on the other side of the equation, which would give the equivalent form
negative 10𝑦 plus three 𝑥 plus 50 equals zero, the exact negative of the equation