Find the 𝑥-coordinate of the point
at which the straight line three 𝑥 plus nine 𝑦 equals zero cuts the 𝑥-axis.
There are lots of ways of
approaching this question. Let’s begin by considering the
𝑥𝑦-plane as shown. Any point which cuts the 𝑥-axis
will have a 𝑦-coordinate equal to zero. This means that we can substitute
𝑦 equals zero into the equation three 𝑥 plus nine 𝑦 equals zero. This gives us three 𝑥 plus nine
multiplied by zero equals zero.
As nine multiplied by zero is zero,
we are left with three 𝑥 is equal to zero. We can then divide both sides of
this equation by three. On the left-hand side the threes
cancel, and on the right-hand side zero divided by three is zero. The 𝑥-coordinate of the point at
which the straight line three 𝑥 plus nine 𝑦 equals zero cuts the 𝑥-axis is
An alternative method would be to
rewrite our equation in slope–intercept form, 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 is
the slope or gradient and 𝑏 is the 𝑦-intercept. Subtracting three 𝑥 from both
sides of the original equation, we have nine 𝑦 is equal to negative three 𝑥. We can then divide both sides of
the equation by nine such that 𝑦 is equal to negative three-ninths 𝑥. As both the numerator and
denominator of the fraction are divisible by three, this can be rewritten as 𝑦 is
equal to negative one-third 𝑥.
The equation three 𝑥 plus nine 𝑦
equals zero has a slope or gradient equal to negative one-third and a 𝑦-intercept
equal to zero. This linear equation can be drawn
on the 𝑥𝑦-plane as shown. As this passes through the origin,
this confirms that the 𝑥-coordinate where the line cuts the 𝑥-axis is zero.