253 2.5 Equations of Lines and Linear Models Point-Slope Form The graph of a linear function is a straight line. We now develop various forms for the equation of a line. Figure 45 shows the line passing through the fixed point 1x1, y12 having slope m. (Assuming that the line has a slope guarantees that it is not vertical.) Let 1x, y2 be any other point on the line. Because the line is not vertical, x - x1 ≠0. Now use the definition of slope. m= y - y1 x - x1 Slope formula m1x - x12 = y - y1 Multiply each side by x - x1. y - y1 = m1x - x12 Interchange sides. This result is the point-slope form of the equation of a line. 2.5 Equations of Lines and Linear Models ■ Point-Slope Form ■ Slope-Intercept Form ■ Vertical and Horizontal Lines ■ Parallel and Perpendicular Lines ■ Modeling Data ■ Graphical Solution of Linear Equations in One Variable 0 x y (x, y) (x1, y1) Fixed point Slope = m Any other point on the line Figure 45 LOOKING AHEAD TO CALCULUS A standard problem in calculus is to find the equation of the line tangent to a curve at a given point. The derivative (see Looking Ahead to Calculus earlier in this chapter) is used to find the slope of the desired line, and then the slope and the given point are used in the point-slope form to solve the problem. Point-Slope Form The point-slope form of the equation of the line with slope m passing through the point 1x1, y12 is given as follows. y −y1 =m1x −x12 EXAMPLE 1 Using the Point-Slope Form (Given a Point and the Slope) Write an equation of the line through the point 1-4, 12 having slope -3. SOLUTION Here x1 = -4, y1 = 1, and m= -3. y - y1 = m1x - x12 Point-slope form y - 1 = -33x - 1-424 x1 = -4, y1 = 1, m= -3 y - 1 = -31x + 42 y - 1 = -3x - 12 Distributive property y = -3x - 11 Add 1. S Now Try Exercise 29. Be careful with signs. EXAMPLE 2 Using the Point-Slope Form (GivenTwo Points) Write an equation of the line through the points 1-3, 22 and 12, -42. Write the result in standard form Ax + By = C. SOLUTION Find the slope first. m= -4 - 2 2 - 1-32 = - 6 5 Definition of slope The slope m is - 6 5 . Either the point 1-3, 22 or the point 12, -42 can be used for 1x1, y12 in the point-slope form. We choose 1-3, 22.
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