216 Chapter 3 Linear Equations in Two Variables Some of the points on the line are (0, 3), (1, 3), (2, 3), and (-4, 3). The only points that this graph contains are points whose y-value is 3. A key characteristic for an ordered pair to be a solution of the equation of this line is that the y-value equals 3. Therefore, the equation of this line is y = 3. The equation y = 3 is equivalent to 0x + y = 3. Any value of x substituted in this equation is multiplied by 0, so the solutions only depend on the value of y being 3. ������������������������ The graph of any equation of the form y = k, where k is a real number, is a horizontal line through k on the y-axis. The point (0, k) is the y-intercept of the graph. Now, we will examine the graph of a vertical line, as follows. (4, 4) (4, 2) (4, 0) (4, –2) 4 2 2 –2 –4 –2 –4 x y Some of the points on the line are (4, 4), (4, 2), (4, 0), and (4, -2). The only points that this graph contains are points whose x-value is 4. A key characteristic for an ordered pair to be a solution of the equation of this line is that the x-value equals 4. Therefore, the equation of this line is x = 4. The equation x = 4 is equivalent to x + 0y = 4. Any value of y substituted in this equation is multiplied by 0, so the solutions only depend on the value of x being 4. ������������������������ The graph of any equation of the form x = h, where h is a real number, is a vertical line through h on the x-axis. The point (h, 0) is the x-intercept of the graph. �� ������������������������ ������������������ Graph each linear equation. 4a. y = 2 4b. x = -3 4c. 2x + 5 = 0 Solutions 4a. The equation y = 2 can be written as 0x + y = 2. Notice that for any x-value chosen, y is 2. So, any ordered pair whose y-coordinate is 2 is a solution. If x = 3, 0, and -1, the ordered pair solutions are x y (x, y) 3 2 (3, 2) 0 2 (0, 2) -1 2 (-1, 2) (0, 2) 2 4 4 –2 –4 –2 –4 x y The graph is a horizontal line with a y-intercept of (0, 2). Note the graph does not have an x-intercept because the y-coordinate is never zero.
hendricks_beginning_algebra_1e_ch1_3
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