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4.19 (a) Prove that any linear equation ax + by = c is the equation of a line, it being assumed that a and b are not both zero. [Hint: Consider separately the cases b = 0 and b = 0.] (b) Prove that any line is the graph of a linear equation. [Hint: Consider separately the cases where the line is vertical and where the line is not vertical.] (c) Prove that two lines a1 x + b1 y = c1 and a2 x + b2 y = c2 are parallel if and only if a1 b2 = a2 b1 (When a1 = 0 and b1 = 0, the latter condition is equivalent to a2 /a1 = b2 /b1 .) 4.20 If the line L has the equation 3x + 2y 4 = 0, prove that a point P(x, y) is above L if and only if 3x + 2y 4 > 0. 4.21 If the line M has the equation 3x 2y 4 = 0, prove that a point P(x, y) is below M if and only if 3x 2y 4 > 0 . 4.22 (a) Use two inequalities to de ne the set of all points above the line 4x + 3y 9 = 0 and to the right of the line x = 1. Draw a diagram indicating the set. (b) GC Use a graphing calculator to check your answer to part (a). 4.23 (a) The leading car rental company, Heart, charges $30 per day and 15 per mile for a car. The second-ranking company, Bird, charges $32 per day and 12 per mile for the same kind of car. If you expect to drive x miles per day, for what values of x would it cost less to rent the car from Heart (b) GC Solve part (a) using a graphing calculator. 4.24 Draw the graphs of the following equations: (a) |x| |y| = 1; (b) y = 1 (x + |x|). 2 (c) GC Use a graphing calculator to solve part (b). 4.25 Prove the following geometric theorems by using coordinates. (a) The gure obtained by joining the midpoints of consecutive sides of any quadrilateral is a parallelogram. (b) The altitudes of any triangle meet at a common point. (c) A parallelogram with perpendicular diagonals is equilateral (a rhombus). (d) If two medians of a triangle are equal, the triangle is isosceles. (e) An angle inscribed in a semicircle is a right angle. [Hint: For parts (a), (b), and (c), choose coordinate systems as in Fig. 4-16.]
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Fig. 4-16
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4.26 Describe geometrically the families of the following lines, where m and b are any real numbers: (a) y = mx + 2 (b) y = 3x + b
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4.27 The x-intercept of a line L is de ned to be the x-coordinate of the unique point where L intersects the x-axis. Thus, it is the number a for which (a, 0) lies on L . (a) Which lines do not have x-intercepts (b) Find the x-intercepts of (i) 2x 3y = 4, (ii) x + 7y = 14, (iii) 5x 13y = 1, (iv) x = 3, (v) y = 1. (c) If a and b are the x-intercept and y-intercept of a line, and a = 0 and b = 0, show that y x + =1 a b is an equation of the line. (d) If (2) is an equation of a line, show that a is the x-intercept and b is the y-intercept. 4.28 In the triangle with vertices A(3, 1), B(2, 7), and C(4, 10) nd the slope-intercept equation of: BC; (b) the median from B to side AC; (c) the perpendicular bisector of side AB. 4.29 Using coordinates, prove that a trapezoid with equal diagonals is isosceles. (a) the altitude from A to side (2)
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