# Question Video: Finding the Equation of a Straight Line Mathematics

Find the equation of the straight line that passes through the point of intersection of the two straight lines π₯ β 8π¦ = 2 and β6π₯ β 8π¦ = 1 and is parallel to the π¦-axis.

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### Video Transcript

Find the equation of the straight line that passes through the point of intersection of the two straight lines π₯ minus eight π¦ equals two and negative six π₯ minus eight π¦ equals one and is parallel to the π¦-axis.

So weβre looking for the equation of this particular straight line. And the first thing we notice that this line is parallel to the π¦-axis. Now we can recall that lines which are parallel to the π¦-axis are vertical lines. And they therefore have equations of the form π₯ equals π, where π is some constant. And π is the value at which these lines cross the π₯-axis.

We also know that the straight line weβre looking for passes through the point of intersection of the two straight lines whose equations weβve been given. So what weβre going to need to do is solve these two equations simultaneously in order to find their point of intersection. Or at least find the π₯-coordinate of their point of intersection. So these are the two equations weβre looking to solve. Theyβre linear simultaneous equations in two variables.

There are a couple of different approaches we could take. Firstly, we could note that both equations have the same coefficient of π¦. They both have negative eight π¦. And so we could use the acronym SSS, standing for same sign subtract, in order to eliminate the π¦-variables. If we subtract equation two from equation one, then we have π₯ minus negative six π₯, which is π₯ plus six π₯ or seven π₯. We have negative eight π¦ minus negative eight π¦. Thatβs negative eight π¦ plus eight π¦. So the π¦s are eliminated. And on the right-hand side, we have two minus one, which is equal to one. We therefore have the equation seven π₯ equals one, and weβve eliminated the π¦-variable. To solve this equation, we just need to divide both sides by seven, giving π₯ equals one-seventh.

An alternative method would be to use the method of substitution. We could rearrange equation one to give negative eight π¦ is equal to two minus π₯. And we can then substitute this expression for negative eight π¦ into equation two. Doing so gives negative six π₯ plus two minus π₯ is equal to one. And now we have an equation in π₯ only.

We can then simplify to give negative seven π₯ plus two equals one. Subtract two from each side to give negative seven π₯ equals negative one, and then divide by negative seven. Giving π₯ equals negative one over negative seven, which is equal to one-seventh. In both cases then, we found that the π₯-coordinate of the point of intersection of these two straight lines is one-seventh.

Now we donβt need to go any further because we donβt need to know the π¦-coordinate of the point of intersection. Remember, we said straight lines parallel to the π¦-axis have equations of the form π₯ equals some constant. So if our line passes through the point where π₯ equals one-seventh, its equation must just be π₯ equals one-seventh.

So by first recalling the general equation of a straight line which is parallel to the π¦-axis and then partially solving the equations of the two straight lines to find the π₯-coordinate of their point of intersection. Weβve found that the equation of the straight line that passes through their point of intersection and is parallel to the π¦-axis is π₯ equals one-seventh.