Video Transcript
Find the equation of the straight line that is parallel to the 𝑦-axis and passes through the point of intersection of the two straight lines 𝑦 equals negative three and 𝑥 equals eleven fifteenths 𝑦.
Sketching in an 𝑥𝑦-plane, where the axes cross at our origin, we can draw in the two straight lines given to us as 𝑦 equals negative three and 𝑥 equals eleven fifteenths 𝑦. 𝑦 equals negative three is a horizontal line that crosses the 𝑦-axis at negative three, while 𝑥 equals eleven fifteenths 𝑦 passes through the origin and has a positive slope like this. The equation of the straight line we want to solve for is parallel to the 𝑦-axis and it passes through the point of intersection of these two lines.
A line that’s parallel to the 𝑦-axis is one that has a constant 𝑥-value. In other words, it would be a vertical line. And we know it passes through our point of intersection. We can completely describe this straight line by writing out the 𝑥-value of all points on the line. That 𝑥-value will be the 𝑥-coordinate of this point of intersection of our two orange lines. Regarding that point, we know its 𝑦-value is negative three because the point lies along the line 𝑦 equals negative three.
To find the corresponding 𝑥-value, we can use the fact that this point of intersection also lies along the line 𝑥 equals eleven fifteenths 𝑦. At the intersection point, we know that 𝑦 equals negative three. So if we substitute that in for 𝑦 in this equation, then 𝑥, we see, equals negative eleven-fifths. The coordinates of our point of intersection then are negative eleven-fifths, negative three. And therefore, this line consists simply of every point in the 𝑥𝑦-plane, where 𝑥 equals negative eleven-fifths. That fact lets us write the equation of this straight line. The equation of the line parallel to the 𝑦-axis and passing through the given point of intersection is 𝑥 equals negative eleven-fifths.
