Find equation of line that is parallel to given line and includes given point.Line is `y = 2x-1` Point is `(1,a)` Also, find equation of a line that is perpendicular to given line through `(1, a)`.
Hey, so, you didn't give us the full point, but I made it general for you, so you just need to plug in your value for "a," and it will work out!
So, let's start by recognizing what it means for 2 lines to be parallel: to never intersect. This means that they are the same distance apart in the x-y axis at each point! What does this mean for the lines, though?
This means the lines must have the same slope. If they have even slightly different slopes, the lines will eventually intersect, and they won't be parallel!
So, let's look at a graph of the line we're supposd to be parallel to:
From this, we can see the basic slope of the line and the slope we'd need to match to solve the problem.
So, let's go back to our basic slope-intercept form for a line:
`y = mx + b`
Where `m` is the slope, and `b` is the y-intercept. Well, we already have one of these from the given equation! `m=2` to make sure our slopes match!
`y = 2x + b`
However, we need to find the y-intercept to complete the equation. To do this, we're going to need to plug in the point we want the line to go through: (1, a)
`a = 2*1 + b`
`a = 2 + b`
So, we just need to solve for b by subtracting 2 from both sides:
`a-2 = b`
So, we have our equation for the parallel line:
`y = 2x + a-2`
So, again, what you'd need to do for your exact problem is replace "a" and you're finished!
Now to find the perpendicular line. The slope here (let's call `m_p`) has an interesting rule associated with the slope of the original line:
`m_p = -1/m`
So, we take the negative reciprocal, and we'll have the slope of the perpendicular! This gives us a new slope of `-1/2`.
So, again, we have our basic slope intercept form:
`y = -1/2 x + b`
Now, we put in the point (again):
`a = -1/2*1 + b`
Solving, we add 1/2 to both sides:
`a+1/2 = b`
And now we have our y-intercept and our complete equation!
`y = -1/2x + a + 1/2`
I hope that helps! Again, just plug in the value you have for "a" and this should work out for the particular problem you have. Pay attention to how it was solved, though, so it makes sense when they ask it later!