Video Transcript
Find the equation of the straight line passing through the point of intersection of the two straight lines 𝑦 minus nine is equal to zero and 𝑥 minus 𝑦 is equal to zero that intersects the positive directions of the coordinate axes in two points at the same distance from the origin.
In this question, we’re asked to find the equation of a straight line, and we’re given some information to help us determine the equation of this line. First, we’re told that this straight line passes through the point of intersection between two other straight lines. And we’re given their equations: 𝑦 minus nine is equal to zero and 𝑥 minus 𝑦 is equal to zero. We’re also told that the straight line we need to find the equation of intersects the positive directions of both coordinate axes in two points at the same distance from the origin. And this is useful information. To help us understand this, let’s sketch what we’ve been given.
We’re told that this straight line intersects both coordinate axes on the positive side. And remember, we usually say the 𝑦-intercept of a straight line is called 𝑏. So let’s call this value 𝑏. But we’re also told in the question that these intercepts are equidistant from the origin. So the 𝑥-intercept must also be at a distance of 𝑏 from the origin. This allows us to sketch the straight line we need to find the equation of. And we can already notice something interesting about this graph. We can determine the slope of this line. We can do this by noting for every 𝑏 units we move to the right, the line moves 𝑏 units down. So the slope of this line is negative one.
It’s also worth noting we could have done this by substituting the points zero, 𝑏 and 𝑏, zero into the equation for the slope of a line. This would have given us negative one times 𝑏 over 𝑏, which is negative one. We’ve now found the slope of this straight line. And we’re told that the line passes through the point of intersection between 𝑦 minus nine is equal to zero and 𝑥 minus 𝑦 is equal to zero. So if we find this point, we can use the point–slope form of the equation of a line to determine the equation of this line. And we can find the coordinates of this point by recalling we can find the coordinates of the point of intersection between two lines by solving them as simultaneous equations. We need to solve the equations 𝑦 minus nine is equal to zero and 𝑥 minus 𝑦 is equal to zero.
There’s a few different ways of doing this. We can let the first equation does not contain the value of 𝑥. So we can just solve this for 𝑦. We just add nine to both sides of the equation. This gives us that 𝑦 is equal to nine. Now we substitute 𝑦 is equal to nine into the second equation. This will allow us to find the 𝑥-coordinate of the point of intersection. This gives us that 𝑥 minus nine is equal to zero. Now we solve for 𝑥 by adding nine to both sides of the equation. This then gives us that 𝑥 is equal to nine. So the coordinates of the point of intersection between these two lines is the point nine, nine. Therefore, the slope of the straight line we need to find the equation of is negative one, and it passes through the point nine, nine.
We can now find the equation of the straight line by using the point–slope form for the equation of a straight line. We recall this tells us if a line has slope 𝑚 and passes through the point with coordinates 𝑥 sub one, 𝑦 sub one, then its equation can be given by 𝑦 minus 𝑦 sub one is equal to 𝑚 times 𝑥 minus 𝑥 sub one. And we’ve already shown the slope of this line is negative one and it passes through the point with coordinates nine, nine. We then substitute these values into the point–slope form. We get 𝑦 minus nine is equal to negative one times 𝑥 minus nine. We can now simplify this equation. We’ll distribute the negative over our parentheses. We get 𝑦 minus nine is equal to negative 𝑥 plus nine.
And finally, we’ll write this in the general form of the equation of a straight line by taking all of the terms onto the left-hand side of the equation. We get that 𝑥 plus 𝑦 minus 18 is equal to zero, which is our final answer. Therefore, we were able to show the equation of the straight line passing through the point of intersection between the two straight lines 𝑦 minus nine is equal to zero and 𝑥 minus 𝑦 is equal to zero which intersects the positive directions of the coordinate axes in two points at the same distance from the origin is 𝑥 plus 𝑦 minus 18 is equal to zero.
