SECTION 1.5 Lines 39 6 Find the Equation of a Line Given Two Points Finding an Equation of a Line Given Two Points Find an equation of the line containing the points ( ) 2, 3 and ( ) −4, 5 . Graph the line. Solution EXAMPLE 7 First compute the slope of the line. = − − − = − = − m 5 3 4 2 2 6 1 3 = − − m y y x x 2 1 2 1 Use the point ( ) 2, 3 and the slope = − m 1 3 to get the point-slope form of the equation of the line. ( ) − = − − y x 3 1 3 2 ( ) − = − y y m x x 1 1 Continue to get the slope-intercept form. − = − + = − + y x y x 3 1 3 2 3 1 3 11 3 = + y mx b See Figure 63 for the graph. Figure 63 =− + y x 1 3 11 3 x y –4 10 (–4, 5) (2, 3) 2 –2 0, 11 –– 3 ( ) In the solution to Example 7, we could have used the other point, ( ) −4, 5 , instead of the point ( ) 2, 3 .The equation that results, when written in slope-intercept form, is the equation that we obtained in the example. (Try it for yourself.) Now Work PROBLEM 45 7 Graph Lines Written in General Form Using Intercepts DEFINITION General Form The equation of a line is in general form * when it is written as + = Ax By C (4) where A , B , and C are real numbers and A and B are not both 0. If = B 0 in equation (4), then ≠ A 0 and the graph of the equation is a vertical line: = x C A . If ≠ B 0 in equation (4), then we can solve the equation for y and write the equation in slope-intercept form as we did in Example 6. One way to graph a line given in general form, equation (4), is to find its intercepts. Remember, the intercepts of the graph of an equation are the points where the graph crosses or touches a coordinate axis. Graphing an Equation in General Form Using Its Intercepts Graph the equation + = x y 2 4 8 by finding its intercepts. Solution EXAMPLE 8 To obtain the x -intercept, let = y 0 in the equation and solve for x . + = + ⋅ = = = x y x x x 2 4 8 2 4 0 8 2 8 4 Let = y 0 . Divide both sides by 2. *Some texts use the term standard form . (continued)
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