Question Video: Finding the Equation of a Line Passing Through the Origin and the Intersection Point of Two Given Lines Mathematics

Video Transcript

The equation of the straight line which passes through the origin point and the intersection point of the two lines 𝑥 minus two equals zero and 𝑦 plus seven equals zero is what. Two 𝑥 plus seven 𝑦 equals zero, two 𝑥 minus seven 𝑦 equals zero, seven 𝑥 plus two 𝑦 equals zero, or seven 𝑥 minus two 𝑦 equals zero.

So we’re looking for the equation of a straight line. We’re told first of all that it passes through the origin point. That’s the point with coordinates zero, zero. We also know that the line passes through the intersection point of the lines 𝑥 minus two equals zero and 𝑦 plus seven equals zero. Now, the equations of each of these lines can be rearranged slightly. By adding two to each side, 𝑥 minus two equals zero becomes 𝑥 equals two. And by subtracting seven from each side, 𝑦 plus seven equals zero becomes 𝑦 equals negative seven. And now we have the equations of these two lines in a slightly more recognizable format.

We should recall that an equation of the form 𝑥 equals constant is a vertical line because every point on the line has the same 𝑥-coordinate, in this case, an 𝑥-coordinate of two. On the other hand, an equation of the form 𝑦 equals constant is a horizontal line. In this case, it’s the line passing through negative seven on the 𝑦-axis. Every point on the line has a 𝑦-coordinate of negative seven. The intersection point of these two lines then will have the same 𝑥-coordinate as all points on the vertical line, that’s two, and the same 𝑦-coordinate as all points on the horizontal line. That’s negative seven.

So we now know that the line we’re looking for passes through the origin and it passes through the point with coordinates two, negative seven. We can use these two points to calculate the slope of the line. The formula for calculating the slope of a line which passes through the two points 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two is 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one, which we can also think of as change in 𝑦 over change in 𝑥. Substituting the values for our two points, we have negative seven minus zero for the change in 𝑦 and two minus zero for the change in 𝑥, which of course simplifies to simply negative seven over two.

Now that we know the slope of our line, we can use the point slope form to find its equation. This is 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one, where 𝑚 represents the slope of the line and 𝑥 one, 𝑦 one represents the coordinates of any point on the line. Substituting negative seven over two for the slope 𝑚 and the simpler point zero, zero for the point 𝑥 one, 𝑦 one, we have 𝑦 minus zero equals negative seven over two multiplied by 𝑥 minus zero, which just simplifies to 𝑦 equals negative seven over two 𝑥. Now this is looking promising because we have sevens and twos involved in each of the four options we were given for the equation of the straight line. But our line isn’t in the same format as any of these, so we need to rearrange.

If we multiply both sides of the equation by two, this will eliminate the fractions on the right-hand side, giving two 𝑦 is equal to negative seven 𝑥. Finally, we want to group all the terms on the same side of the equation, which we can do by adding seven 𝑥 to each side. Doing so gives seven 𝑥 plus two 𝑦 is equal to zero. And if we look carefully at the four options we were given, we can see that’s this option here. So we found that the equation of the straight line which passes through the origin and the intersection point of the two lines 𝑥 minus two equals zero and 𝑦 plus seven equals zero is seven 𝑥 plus two 𝑦 is equal to zero.