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Equation of a Line Department of Mathematics University of Leicester

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Contents Using the EquationIntroductionForming the Equation Parallel and Perpendicular Lines

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Introduction We are used to the equation for a straight line. We’ll look at where this equation comes form, how to form it and what we can do with it once we’ve got it. We are also going to see alternative ways of writing the same equation. Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Forming the Equation Suppose we have: How do we get the y-value from the x-value? It’s a straight line, so the ratio between x and y values is the same for all values. Eg. For this line,, or. Ratio = = gradient, or m. We get. Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines See what happens if you vary the value of m.

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Forming the Equation What if the line doesn’t go through the origin? If the line starts at : Then it’s, but with c added on to all the y-values. So. Next c Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines See what happens if you vary the value of c.

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But if we’re not starting at anymore, we can’t just find by finding the ratio of the values. So instead, we use the ratio of the difference in value: Forming the Equation Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Forming the Equation If the line slopes down instead of up, the change in x will be negative: So the gradient will be negative. Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines Try plotting an equation

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What is the equation of this line: Forming the Equation Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Forming the Equation What is the equation of this line: Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Question: What is the equation of the line joining and ? Answer: Start with, Then we know, so, so, or Forming the Equation: Alternative ways of writing it Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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You could rewrite with everything on the same side of the equation: And then if we have fractions, multiply through by the denominators to get Eg. Forming the Equation: Alternative ways of writing it Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Which of the following lines joins and ? Forming the Equation: Alternative ways of writing it All of them Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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If two lines are not parallel then they will have exactly one point of intersection. You can find this by letting the 2 lines have the same - and -values, so they become 2 simultaneous equations… eg. The point of intersection is (-3,-3). Using the equation: Intersection Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Solve the simultaneous equations: Where do and intersect? Using the equation: Intersection Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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You can find the distance between two points using Pythagoras’s theorem: eg. Find the distance between (1,-2) and (3,1). We get: Then distance = hypotenuse=. Using the equation: Distance Next (1,-2) (3,1) Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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What is the distance between the points (3,6) and (-4,10)? Using the equation: Distance Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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To find the midpoint of two points, you just find the average of the -coordinate and the average of the -coordinate. So the midpoint of and is: Using the equation: Midpoint Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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What is the midpoint of (5,7) and (-3,11)? Using the equation: Midpoint Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Parallel lines will have the same gradient, because for the same change in, they both have the same change in. So. Parallel and Perpendicular Lines Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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For perpendicular lines,, and. So. Parallel and Perpendicular Lines Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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What is the gradient of the line parallel to ? Parallel and Perpendicular Lines Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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Which of the following lines is perpendicular to ? Parallel and Perpendicular Lines Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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A straight line has the equation:, or m = gradient, c = y-intercept We can find the Intersection Point of 2 lines, and also the Distance or Midpoint between 2 points. For parallel lines,. For perpendicular lines,. Conclusion Next Using the Equation Introduction Forming the Equation Parallel and Perpendicular Lines

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