Graphing Slanted LinesDate: 07/22/97 at 16:05:47 From: P. J. Richardson Subject: Graphing slanted lines When graphing slanted lines what does it mean when the instructions say add, subtract, or divide both sides by a particular number? And how do you know when to add, subtract, or divide? Please give an example. Regards, P.J. Date: 07/28/97 at 12:22:03 From: Doctor Rob Subject: Re: Graphing slanted lines Suppose you had the equation 2*x - 3*y + 5 = 0. Since it has degree 1, it is the equation of a line. As an equation, it has two sides: the left, 2*x - 3*y + 5, and the right, 0. You are going to try to solve this equation for y. The way to do that is to move terms from one side to the other by adding something to both sides. For example, to move the 5 from the left side to the right, add -5 to both sides. Since -5 = -5, and equals added to equals are equal, you get (2*x - 3*y + 5) + (-5) = 0 + (-5) or, in other words, 2*x - 3*y = -5. Move the 2*x term to the other side by adding -2*x = -2*x to this last equation: 2*x - 3*y + (-2*x) = -5 + (-2*x), or -3*y = -2*x - 5. We have isolated y on one side of the equation, and everything else on the other side, which is desirable. Now we are going to divide this equation by -3 = -3. Since equals divided by nonzero equals are equal, we will get a true equation: (-3*y)/(-3) = (-2*x - 5)/(-3) or y = (2/3)*x + (5/3). This completes the process of solving for y. We solved for y to put the equation into the slope-intercept form, y = m*x + b. In this form, y = m*x + b, you can read off m = 2/3 and b = 5/3. m is the slope of the line, and b is the y-intercept, that is, the point (0,b) on the line and also on the y-axis. Now you can graph the line quite easily. Another method is to rewrite the equation in the form x/a + y/b = 1. To do this, move all the constant terms to the right side, and all the x- and y-terms to the left side. Divide both sides by the constant on the right to make it equal to 1. Gather like terms, then set a to be the reciprocal of the x-coefficient and b the reciprocal of the y-coefficient. In this case, we are part way there with the equation 2*x - 3*y = -5. Divide both sides by -5 (i.e. divide by the equation -5 = -5). Then (2*x - 3*y)/(-5) = (-5)/(-5) or (-2/5)*x + (3/5)*y = 1 or x/(-5/2) + y/(5/3) = 1. The right side is 1 as desired for this form of the equation of the line. From this form you can read off a = -5/2 and b = 5/3. These are the x-intercept (a,0) and y-intercept (0,b) of the line. From these two points it is easy to draw the line. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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