Solving Systems of Equations Graphically and AlgebraicallyDate: 05/01/98 at 21:55:17 From: Audra Crystal Patterson Subject: Algebra one(Graphing Systems of Equations) I really do not understand how to work these problems: y = -x y = 2x I have to graph each system of equations, then tell whether the system has one solution, no solution, or infinitely many solutions. If the system has one solution, name it. If you could please try and tell me and give examples, maybe it would help me understand them better. Thank you. Audra Crystal Patterson Date: 05/04/98 at 11:27:16 From: Doctor Loni Subject: Re: Algebra one (Graphing Systems of Equations) Let's see if I can help you at least a little bit. I will have to make a few assumptions about what you have learned. (I will let you know what I am assuming as we go along.) Each of the equations in your example is a linear equation; in other words, the equation describes a line. I am going to assume you know how to graph points and that you know a little bit about lines. If the two lines cross at one point, there is one solution to the equations, and the solution is the x and y values of the point where they cross (or intersect). If the two lines are parallel (the two yellow lines down a highway are parallel lines -- they are always the same distance apart), there is no solution, because they will never cross. There are infinitely many solutions if the two equations define the same line, which means they have every point in common. Lines can be written in a form like this: y = mx + b where m and b are constants. m is the slope of the line, and b is the point where the line crosses the y axis, also known as the y- intercept. Just like the slope of a hill, the slope of a line is the degree of slant. If two lines are parallel to each other, they will have the same slope. So if you have two equations like this: y = 2x + 6 and y = 2x + 1 there will be no solution, because they have the same slope and will never cross. If two lines are not parallel, there are two ways to find the point at which they intersect (which is the solution to the two equations): 1) You can do it graphically. If your equation is y = 3x, you could put in values for x, solve for y, and then graph the points. For instance, if x is 2, y would equal 6. You could graph that point. Then pick at least one other value for x, find y, and graph that point. Connect the dots, and you have a graph of your line. Then you would graph the second equation in the same way, and where they intersect would be the solution. 2) You can also find the point of intersection algebraically. For example, say your two equations are: y = 2x + 6 and y = -4x You can use the substitution method, where you solve one equation for one variable, and plug in what you get into the other equation. For instance, you already have: y = -4x You can put -4x in for y in the other equation: y = 2x + 6 - 4x = 2x + 6 Solve for x: x = -1 Now plug your value for x into one of the equations and solve for y: y = -4x y = -4(-1) = 4 So the solution to the two equations is x = -1 and y = 4; or, written as an ordered pair, (-1,4). That is the point where those two lines would intersect. Hope this is helpful. Let us know if you need more clarification. -Doctor Loni, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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