40 CHAPTER 1 Graphs Figure 64 + = x y 2 4 8 x y –3 3 (0, 2) (4, 0) 4 The x -intercept is 4, and the point ( ) 4, 0 is on the graph of the equation. To obtain the y -intercept, let = x 0 in the equation and solve for y . + = ⋅ + = = = x y y y y 2 4 8 2 0 4 8 4 8 2 The y -intercept is 2, and the point ( ) 0, 2 is on the graph of the equation. Plot the points ( ) 4, 0 and ( ) 0, 2 and draw the line through the points. See Figure 64. Let =x 0. Divide both sides by 4. Now Work PROBLEM 99 The equation of every line can be written in general form. For example, a vertical line whose equation is = x a can be written in the general form ⋅ + ⋅ = x y a 1 0 = = = A B C a 1, 0, A horizontal line whose equation is = y b can be written in the general form ⋅ + ⋅ = x y b 0 1 = = = A B C b 0, 1, Lines that are neither vertical nor horizontal have general equations of the form + = ≠ ≠ Ax By C A B 0 and 0 Because the equation of every line can be written in general form, any equation equivalent to equation (4) is called a linear equation . THEOREM Criteria for Parallel Lines Two nonvertical lines are parallel if and only if their slopes are equal and they have different y -intercepts. The use of the phrase “if and only if” in the preceding theorem means that actually two statements are being made, one the converse of the other. • If two nonvertical lines are parallel, then their slopes are equal and they have different y -intercepts. • If two nonvertical lines have equal slopes and they have different y -intercepts, then they are parallel. Figure 65 Parallel lines y Rise Run Run Rise x 8 Find Equations of Parallel Lines When two lines (in the plane) do not intersect (that is, they have no points in common), they are parallel . Look at Figure 65.We have drawn two parallel lines and have constructed two right triangles by drawing sides parallel to the coordinate axes. The right triangles are similar. (Do you see why? Two angles are equal.) Because the triangles are similar, the ratios of corresponding sides are equal.
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