First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow. Show The PointsWe use Cartesian Coordinates to mark a point on a graph by how far along and how far up it is: Example: The point (12,5) is 12 units along, and 5 units up StepsThere are 3 steps to find the Equation of the Straight Line :
Step 1: Find the Slope (or Gradient) from 2 PointsWhat is the slope (or gradient) of this line? We know two points:
The slope is the change in height divided by the change in horizontal distance. Looking at this diagram ... Slope m = change in ychange in x = yA − yBxA − xB In other words, we:
Like this: m = change in y change in x = 4−3 6−2 = 1 4 = 0.25 It doesn't matter which point comes first, it still works out the same. Try swapping the points: m = change in y change in x = 3−4 2−6 = −1 −4 = 0.25 Same answer. Step 2: The "Point-Slope Formula"Now put that slope and one point into the "Point-Slope Formula" Start with the "point-slope" formula (x1 and y1 are the coordinates of a point on the line): y − y1 = m(x − x1) We can choose any point on the line for x1 and y1, so let's just use point (2,3): y − 3 = m(x − 2) We already calculated the slope "m": m = change in ychange in x = 4−36−2 = 14 And we have: y − 3 = 14(x − 2) That is an answer, but we can simplify it further. Step 3: SimplifyStart with:y − 3 = 14(x − 2) Multiply 14 and (x−2):y − 3 = x4 − 24 Add 3 to both sides:y = x4 − 24 + 3 Simplify:y = x4 + 52 And we get: y = x4 + 52 Which is now in the Slope-Intercept (y = mx + b) form. Check It!Let us confirm by testing with the second point (6,4): y = x/4 + 5/2 = 6/4 + 2.5 = 1.5 + 2.5 = 4 Yes, when x=6 then y=4, so it works! Another ExampleExample: What is the equation of this line?Start with the "point-slope" formula: y − y1 = m(x − x1) Put in these values:
And we get: y − 6 = −2(x − 1) Simplify to Slope-Intercept (y = mx + b) form: y − 6 = −2x + 2 y = −2x + 8 DONE! The Big ExceptionThe previous method works nicely except for one particular case: a vertical line: A vertical line's gradient is undefined (because we cannot divide by 0): m = yA − yBxA − xB = 4 − 12 − 2 = 30 = undefined But there is still a way of writing the equation: use x= instead of y=, like this: x = 2 What is the equation of a line when given two points?The two-point form of a line is used for finding the equation of a line given two points (x1,y1) ( x 1 , y 1 ) and (x2,y2) ( x 2 , y 2 ) on it. The two point-form of a line is:y−y1=y2−y1x2−x1(x−x1) y − y 1 = y 2 − y 1 x 2 − x 1 ( x − x 1 ) OR y−y2=y2−y1x2−x1(x−x2) y − y 2 = y 2 − y 1 x 2 − x 1 ( x − x 2 ) .
How do you find the equation of a line with 5 points?First, you should plug the given points, (5, –8) (–2, 6), into the slope formula to find the slope of the line. Then, plug the slope into the slope formula, y = mx + b, where m is the slope. Plug in either one of the given points, (5, –8) or (–2, 6), into the equation to find the y-intercept (b).
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