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Exmple 214 Find the equation of the straight lines with the given points. a d b e c f

Exmple 214214 Find the equation of the straight lines with the given points.\newlinea\newlined\newlineb\newlinee\newlinec\newlinef

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Q. Exmple 214214 Find the equation of the straight lines with the given points.\newlinea\newlined\newlineb\newlinee\newlinec\newlinef
  1. Calculate Slope: To find the equation of a line, we need the slope mm which is the change in yy over the change in xx. So, we calculate the slope using the points (a,d)(a, d) and (b,e)(b, e).m=e−db−am = \frac{e - d}{b - a}
  2. Point-Slope Form: Now we use the point-slope form of the equation of a line, which is y−y1=m(x−x1)y - y_1 = m(x - x_1), where (x1,y1)(x_1, y_1) is a point on the line and mm is the slope.\newlineWe can use point (a,d)(a, d) for this.\newliney−d=m(x−a)y - d = m(x - a)
  3. Substitute Slope: Substitute the value of mm from step 11 into the equation from step 22.y−d=(e−d)(b−a)(x−a)y - d = \frac{(e - d)}{(b - a)}(x - a)
  4. Simplify Equation: Now we simplify the equation by distributing the slope on the right side of the equation. \newliney−d=e−db−a⋅x−e−db−a⋅ay - d = \frac{e - d}{b - a} \cdot x - \frac{e - d}{b - a} \cdot a
  5. Isolate y: Next, we simplify the equation further by multiplying out the terms on the right side.\newliney−d=e−db−a⋅x−e−db−a⋅ay - d = \frac{e - d}{b - a} \cdot x - \frac{e - d}{b - a} \cdot a
  6. Slope-Intercept Form: To write the equation in slope-intercept form y=mx+by = mx + b, we need to isolate yy.y=e−db−a⋅x−(e−d)⋅ab−a+dy = \frac{e - d}{b - a} \cdot x - \frac{(e - d) \cdot a}{b - a} + d
  7. Standard Form: Finally, we simplify the equation by combining like terms and writing it in the standard form, which is Ax+By=CAx + By = C. Let's multiply everything by (b−a)(b - a) to get rid of the fraction. $(b - a)y = (e - d)x - (e - d)a + (b - a)d
  8. Expand and Simplify: Now we expand and simplify the equation.\(\newline\)\((b - a)y = (e - d)x - (e - d)a + bd - ad\)

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