Question Video: Using the Equation of a Straight Line to Find the 𝑥-Coordinate of a Point

Video Transcript

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 zero.

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.