Find the equation of the straight line that passes through the origin and the point of intersection of the two straight lines 𝑥 equals negative seventeen-fourths and 𝑦 equals negative five.
If we sketch in an 𝑥𝑦-coordinate plane, then we can draw in this given straight line 𝑥 equals negative seventeen-fourths — this is a vertical line crossing the 𝑥-axis at negative seventeen-fourths — and also the straight line 𝑦 equals negative five — that would be a horizontal line that crosses the 𝑦-axis at negative five. The straight line whose equation we want to solve for passes through the point of intersection of these two lines, and it also passes through the origin. So it’s this pink line then whose equation we want to find.
As we think of the equation of a straight line, there are different ways of writing this expression. The way we’ll use is to say that the 𝑦-coordinate of the line is equal to the slope 𝑚 of the line multiplied by the 𝑥-value plus this value 𝑏, which is the 𝑦-intercept of the line, the place where it crosses the 𝑦-axis. If 𝑦 equals 𝑚𝑥 plus 𝑏, it’s 𝑚 and 𝑏 that we want to find. And let’s start by solving for the slope 𝑚 of this pink line here.
The slope 𝑚 is equal to the difference in the 𝑦-coordinates of two points on a given line divided by the difference in the 𝑥-coordinates of those corresponding points. In our scenario, two points on the line are first this point of intersection with coordinates negative seventeen-fourths, negative five and then second the origin with coordinates zero, zero.
If we consider these points from left to right so that this point of intersection is our first point and the origin is our second, then we write that 𝑦 two, that’s zero, minus 𝑦 one, that’s negative five, divided by 𝑥 two, that’s also zero, minus 𝑥 one, that’s negative seventeen-fourths, is equal to the slope 𝑚. This equals five over seventeen-fourths or, if we multiply top and bottom by four, twenty seventeenths. This is the slope 𝑚 of our line.
We can update our expression then for the equation of our straight line. It’s 𝑦 equals twenty seventeenths 𝑥 plus 𝑏. Like we said, this value 𝑏 is the 𝑦-intercept of the line. And we notice now that since this line passes through the origin, the 𝑦-intercept is zero, that point at the origin. This tells us that 𝑏 itself is zero for our straight line. And so, our equation simplifies to 𝑦 equals twenty seventeenths 𝑥. This is the equation of the straight line passing through these two points.