Video Transcript
Determine the equation of the line passing through the point of intersection of the two lines whose equations are five 𝑥 plus two 𝑦 is equal to zero and three 𝑥 plus seven 𝑦 plus 13 is equal to zero, while making an angle of 135 degrees with the positive 𝑦-axis.
In this question, we’re asked to find the equation of a line. And we’re told that this line passes through the points of intersection between two other straight lines: the line five 𝑥 plus two 𝑦 is equal to zero and the line three 𝑥 plus seven 𝑦 plus 13 is equal to zero. And we’re also told that this line makes an angle of 135 degrees with the positive 𝑦-axis. To answer this question, let’s start by using the information we’re given. We’re going to start by using the fact that our line makes an angle of 135 degrees with the positive 𝑦-axis. And to help us use this information, let’s start by sketching our line and the angle onto a diagram.
First, we need to measure our angle from the positive direction of the 𝑦-axis. And remember, this angle is positive, so it’s measured counterclockwise. This gives us a sketch which looks somewhat like the following. And there’s something interesting we can notice about this angle. It will allow us to find the angle that this line makes with the positive direction of the 𝑥-axis. First, the 𝑦-axis is a straight line, so 135 degrees plus this angle must be 180 degrees. This must be an angle of 45 degrees. Next, the 𝑥- and 𝑦-axis meet at right angles.
Finally, the sum of the measures of the internal angles in a triangle is 180 degrees. So the remaining angle in this right triangle is 45 degrees. And this is the angle that our line makes with the positive direction of the 𝑥-axis. We can then recall we can use this angle to determine the slope of the line. We know if a line makes an angle of 𝜃 with the positive direction of the 𝑥-axis, then the slope of this line will be equal to the tan of 𝜃. Well, it’s worth pointing out if 𝜃 is 90 degrees, then we get the tan of 90 degrees, which is undefined. However, this will only happen if we have a vertical line. We can apply this to find the slope 𝑚 of this line. The angle it makes with the positive direction of the 𝑥-axis is 45 degrees. So 𝑚 is equal to the tan of 45 degrees, which we know is equal to one.
We now have the slope of our straight line. And we’re told that the line passes through the point of intersection between two given straight lines. So if we find the coordinates of this point of intersection, we can find the point–slope form of the line. And we can find the coordinates of the point of intersection between two lines by recalling the 𝑥- and 𝑦-coordinates of the points of intersection must satisfy both equations of the lines. In other words, we need to solve these equations as a pair of simultaneous equations.
There’s a few different ways of doing this. For example, we could use elimination. However, we’re going to use substitution. We can rearrange the first equation to make 𝑦 the subject. We subtract five 𝑥 from both sides of the equation and then divide through by two. We get that 𝑦 is equal to negative five over two 𝑥. We can now substitute this expression for 𝑦 into the second equation. This then gives us that three 𝑥 plus seven times negative five over two 𝑥 plus 13 is equal to zero.
We now want to solve this equation for 𝑥, so let’s start by simplifying the equation. First, seven times negative five over two is negative 35 over two. And we can subtract 13 from both sides of the equation to get three 𝑥 minus 35 over two 𝑥 is equal to negative 13. Next, we need to simplify the left term. We can do this by noting three is six over two. We can then simplify the 𝑥-terms. Six minus 35 is negative 29. So we get negative 29 over two times 𝑥 is equal to negative 13.
We can now solve for 𝑥 by dividing both sides of the equation through by negative 29 over two. This is, of course, the same as multiplying both sides of the equation through by negative two over 29. We get 𝑥 is negative 13 times negative two over 29. And if we evaluate this, we see it’s equal to 26 over 29. This means we’ve now found the 𝑥-coordinate of the point of intersection between the two lines. We now need to find the 𝑦-coordinate. We can do this by substituting this 𝑥-value into either of the two original simultaneous equations.
However, it’s easier to substitute this value into the rearrangement of the first equation. This gives us that the 𝑦-coordinate of the point of intersection between the two lines is negative five over two multiplied by 26 over 29. We can simplify this equation by canceling the shared factor of two in the numerator and denominator. We get negative five times 13 over 29, which we can evaluate to give us negative 65 over 29. So our line passes through the point with coordinates 26 over 29, negative 65 over 29, and it has a slope of one. This means we can find the equation of this line by using the point–slope form of the equation of a line.
We recall this tells us that a line with a slope of 𝑚 passing through the point with coordinates 𝑥 sub one, 𝑦 sub one will have an equation of 𝑦 minus 𝑦 sub one is equal to 𝑚 times 𝑥 minus 𝑥 sub one. And we’ve already shown that our line has a slope of 𝑚 equal to one and passes through the point with coordinates 26 over 29, negative 65 over 29. So we can substitute these values into the point–slope form of the equation of a line to determine its equation. We get 𝑦 minus negative 65 over 29 is equal to one times 𝑥 minus 26 over 29.
And we can simplify this equation. First, multiplying by one doesn’t change the value, and subtracting negative 65 over 29 is the same as adding 65 over 29. This gives us 𝑦 plus 65 over 29 is equal to 𝑥 minus 26 over 29. We can simplify this equation further by removing the denominators. We’ll do this by multiplying both sides of the equation through by 29. This gives us 29𝑦 plus 65 is equal to 29𝑥 minus 26.
And we’ll now write this in the general form of the equation of a straight line. We’ll do this by subtracting 29𝑦 from both sides of the equation and 65 from both sides of the equation. And since negative 26 minus 65 is equal to negative 91, this gives us 29𝑥 minus 29𝑦 minus 91 is equal to zero, which is our final answer.
