Question Video: Using the Equation of a Straight Line to Find the π‘₯-Coordinate of a Point

Find the π‘₯-coordinate of the point at which the straight line 3π‘₯ + 9𝑦 = 0 cuts the π‘₯-axis.

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Video Transcript

Find the π‘₯-coordinate of the point at which the straight line three π‘₯ plus nine 𝑦 equals zero cuts the π‘₯-axis.

There are lots of ways of approaching this question. Let’s begin by considering the π‘₯𝑦-plane as shown. Any point which cuts the π‘₯-axis will have a 𝑦-coordinate equal to zero. This means that we can substitute 𝑦 equals zero into the equation three π‘₯ plus nine 𝑦 equals zero. This gives us three π‘₯ plus nine multiplied by zero equals zero.

As nine multiplied by zero is zero, we are left with three π‘₯ 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 π‘₯-coordinate of the point at which the straight line three π‘₯ plus nine 𝑦 equals zero cuts the π‘₯-axis is zero.

An alternative method would be to rewrite our equation in slope–intercept form, 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the slope or gradient and 𝑏 is the 𝑦-intercept. Subtracting three π‘₯ from both sides of the original equation, we have nine 𝑦 is equal to negative three π‘₯. We can then divide both sides of the equation by nine such that 𝑦 is equal to negative three-ninths π‘₯. 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 π‘₯.

The equation three π‘₯ 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 π‘₯𝑦-plane as shown. As this passes through the origin, this confirms that the π‘₯-coordinate where the line cuts the π‘₯-axis is zero.

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