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Determine an equation for the line (a) parallel (b) perpendicular to $3 x+7 y=11$ and passing through the point $(1,-3)$.

(a) $3 x+7 y=-18$(b) $7 x-3 y=16$

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

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01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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Finding Parallel and Perpe…

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for this problem, we're gonna find two different equations for lines. One line will be parallel to a given line. One line will be perpendicular. Okay, For both of these lines, I'm comparing it to the same line. So I'm comparing these 23 x plus seven y equals 11. So that's the line I'm comparing Thio. And in both cases, I also wanted to go through the same 0.1 negative three. So what do we mean when we talk about parallel versus perpendicular lines? Well, parallel lines are lines that go next to each other. They will never meet. They never touch. So that means that both of them have to be rising and falling in exactly the same rate. The rate at which align, rises or falls is called the slope. So parallel lines have equal slopes. So if I can find the slope of the equation that I'm comparing this to, I'll know the slope for the parallel line. They'll be the same perpendicular lines cross. In fact, they cross in a very precise way. They make a right angle that's supposed to be a right angle there. Let me make that marking just little bit clearer. Okay, make make a right angle. Now, these obviously do not have equal slopes because they cross each other, but their slopes are related. Perpendicular lines have slopes that are negative, reciprocal of each other. So, for example, if one has a slope of 3/5 the other one will be negative. Five thirds negative, reciprocal. So, again, if I find the slope of my original line, I'll know the slopes of both parallel and perpendicular lines. So let's take our line. We're gonna put this into slope intercept form, which means we solve for why? So I'm gonna move the negative three x over two or the three X over to the right hand side. It becomes negative three x plus 11, and I'm going to divide both sides by seven. So I get why equals negative 37 X plus 11 7th. Okay, slope intercept form. So my slope is the coefficient of my ex term. So my parallel line will have the same slope here. The slope will be negative. Three sevens. The perpendicular line will have negative reciprocal slope, which is a positive seven thirds. Now I have my slopes. I have a point in fact, they're both going to share the same point. So I confined my equations. Let's start with the first one. We'll call that parallel line will call that a So let's start with a I have a point and a slope. So let's use the point slope form. Why? Minus why one equals m times X minus X one. So x one y one is my point. M is my slip. When I go to plug these in my why coordinate is negative three. So why minus negative three or why? Plus three I am. Well, my slope is negative. 3/7 X minus. My ex co Uh, coordinate, which is one. And now I'm just gonna make this look nicer. I'm gonna get rid of my parentheses. That's negative. 37 X plus 3/7. I'm going to subtract three from both sides. When I dio my three on the left hand side is gone. When I subtract three, I'm going to give myself a common denominator instead of three is gonna be 21 7th. So that means my final equation is why equals negative 37 x minus 18 7th. So that's our first equation. The parallel line. So what about be we'll call this perpendicular line be well, we can use the same formula that point slope form. We just have a new slope. Why minus Why won the Y coordinate hasn't changed. New Slope seven thirds and my X coordinate from my point is hasn't changed because that's the same point. That's still gonna be X minus one in the parentheses. And just like last time, let's clean this up. I'll get rid of my parentheses and I'm going to subtract three from both sides. So again, subtracting three removes it from the left. If I subtract three here again, common denominator. Some subtracting nine thirds this time, so I have y equals seven thirds X minus 16 3rd. So those are my two, uh, equations for my lines. The first one is parallel. The second one is perpendicular

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